Bandobras Took
Emperor
I wrote this down to get my thoughts organized, then thought I'd post it here. 
Beginner's Guide to Civ 5 Specialists and Great People
All right. Specialists and Great People were a core element of Civ 4, but they operate somewhat differently Civ 5, from the absence of Priests to the insane abilities of the Great Scientist. This guide will cover the basics of each type of specialist and what they actually do for you, at least until the next patch.
Specialist Basics:
To run a specialist, you need two things:
1) A building which enables a specialist slot; and
2) Enough food to feed the specialist.
Fortunately, Civ 5 gives us two early buildings with +2 food: enough to feed a specialist!
These are the Granary and the Waterwheel. These buildings can be seen as a growth helper, but it can be useful to look on them as specialist enablers -- building one allows you to assign a specialist without the city growing any more slowly than it would without assigning the specialist.
Of the two buildings, the Granary is the better deal, especially with the patch now given bonus food for wheat, bananas, and/or Deer, while a Waterwheel is something you might put in a dedicated GP farm, where you'll assign multiple specialists. Since you need to be by a river for both the Waterwheel and the Gardens (which gives a Great Person increase), this is a natural fit.
Later in the game, the Hospital gives +5 food, and later in the game is when you'll begin unlocking multiple specialist slots of a single type, so this building is a welcome late-game addition.
Specialists put Great Person points towards a great person of their type. Whenever that type's bar touches full, you get a person of that type, that bar is reset to zero, and the amount needed to fill all bars increases. This happily does away with 4 Great Artists spawning off their 1% chance à la Civ 4.
There are some social policies and wonders meant for specialists in the game, as well as one building:
Wonders
Statue of Liberty -- Replacable Parts -- +1 Production/Specialist
Hagia Sophia -- Theology -- +33% Great Person generation in all cities
National Epic -- Philosophy -- Requires Monuments in all cities (which you were probably going to get anyway) +25% Great Person generation in that city
Social Policies
Secularism -- Rationalism -- +2 Science to each specialist
Freedom -- Freedom -- Specialists are 1/2 as unhappy
Civil Society -- Freedom -- Specialists only consume 1 food
Democracy -- Freedom -- Great Person Growth increased by 50%
Meritocracy -- Liberty -- Free Great Person of your choice!
Building
Garden -- Theology -- +25% Great Person generation rate
Each Great Person has a special ability and also a terrain improvement they can be used for. All great people can start golden ages, amplifying production and gold income.
Different Specialists are used for different reasons and in different circumstances as follows:
Scientist:
Enabling Buildings:
University -- Education -- 2
Public School -- Scientific Theory -- 1
Research Lab -- Plastics -- 2
Scientists give +3 Research. Since base research is equal to population, Scientists are worth 3 population points in this regard -- without costing you the unhappiness of 3 population. Run scientists if you want to blow through the tech tree like nobody's business. This ability opens up mid-game with Universities.
Great Scientist:
Special Ability: Great Scientists can automatically discover a tech that is available to research. Research cost is irrelevant. Patching has made Great Scientists rarer; they're now available starting early-mid game, toning down some of the utility of the tech bulb. Nevertheless, always take this option with a Great Scientist; it simply surpasses the others.
Terrain Improvement: +6 research to the tile. The only time it's likely to be worth it to do this is if you have an extremely early Great Scientist (Meritocracy) and are planting them down in your National College city. Otherwise, researching a tech is going to yield more overall beakers, especially in the late game.
Artist:
Enabling Buildings:
Temple -- Philosophy -- 1
Museum -- Archaeology -- 2
Opera House -- Acoustics -- 1
Artists are almost the complete opposite of Scientists. They give +3 culture (HUGELY boosted by the current patch), which is used for expanding workable tiles and gaining social policies. Thus, artists find their best use in a loosely packed empire intending to grow each city to a large size, while Scientists find their best value in small, tightly packed cities. Unlike Science buildings, most culture buildings provide a flat culture rate rather than a percentage increase. Artists give you culture without the corresponding maintenance cost that culture buildings entail.
Great Artist:
Special Ability: Great Artists can perform a culture bomb -- if standing on a tile you own (or a neutral adjacent tile, according to posters below), they can convert every single tile around them to your side, unless the tile in question is an opposing city. This is the only way in the game to take a tile that another Civilization has claimed without war (though you'll take a diplo hit). The best targets for this are city-states, because they won't declare war on you (just get angry) and you won't have to keep shelling out gold to get their luxury resources, but if you want to seize control of a tile from a rival and aren't ready for a war, don't forget this option. There is a 10-turn minimum between Culture Bomb activation.
Terrain Improvement: +6 culture to the tile. This can actually be a worthy tile improvement if you're focusing on culture generation. This is the equivalent of 2 artists for less food investment, and the equivalent of three monuments without the maintenance or increased number of cities. It would be extremely good if there were more culture multipliers, but only the Hermitage and the Broadcast Tower multiply culture rather than providing base culture.
Merchant:
Enabling Buildings:
Market -- Currency -- 1
Bank -- Banking -- 1
Stock Exchange -- Electricity -- 2
Merchants saw a large boost in the last patch. The still generate 2 gold -- but that is no longer as much as a Trade Post, it's twice what a Trade Post (non-riverside) can bring in. Until you reach Economics, running a Merchant instead of working a Trading Post makes sense now.
Great Merchant:
Special Ability: Trade Missions provide a tidy bundle of gold and improved relations with the City-State the Great Merchant has visited. This suggests some synergy with Patronage if going that route.
Terrain Improvement: Customs House; this is essentially two Trade Posts on one tile. +4 gold. Since gold-enhancing buildings are all multipliers, this can actually be a good improvement -- if you will get more gold by working the Trade Post than you would have gotten from a Trade Mission. Both of these are much stronger now that the Trade Post tile improvement has been toned down, especially early on.
Engineer:
Enabling Buildings:
Workshop -- Metal Casting -- 1
Windmill -- Economics -- 1
Factory -- Steam Power -- 2
Engineers generate 2 Hammers, which is okay given that the patch has added more production in many tiles. Engineers are the rarest type of specialist in the game (in that their main enabling building, the Factory, requires a strategic resource to build), but the hammers can be run through some good multipliers.
Great Engineers:
Special Ability: Rush a Wonder. If you really need a wonder quickly, this will aid you, but otherwise, I'd actually give it a miss. It's a temporary production spike, but turn-to-turn production is far more versatile, which leads us to:
Terrain Improvement: +4 Hammers. Assuming a Grassland tile, that's 4 base hammers that feed themselves, which is overall the best production tile in the game. A Riverside Grassland gets even better with a Hydro Plant. If you're planning a super-production city, look to generate Great Engineers and plop them down as tile improvements. Even though production can be a bit more abundant with the new patch, several late-game buildings have actually had their costs increased, which means you'll want to make sure you've still got a production powerhouse at the end of the game.
As with Scientists and Merchants, though, the later in the game, the more likely the insta-build/tech/gold is to be the right choice, as you don't see as much return on the tile improvement investment.
Overall, with the new patch, specialists are evening out, though Great Scientists still win an award as the strongest Great Person type. You can even reasonably set up a GP Farm with the bonus food to the Granary. Specialize away!

Beginner's Guide to Civ 5 Specialists and Great People
All right. Specialists and Great People were a core element of Civ 4, but they operate somewhat differently Civ 5, from the absence of Priests to the insane abilities of the Great Scientist. This guide will cover the basics of each type of specialist and what they actually do for you, at least until the next patch.

Specialist Basics:
To run a specialist, you need two things:
1) A building which enables a specialist slot; and
2) Enough food to feed the specialist.
Fortunately, Civ 5 gives us two early buildings with +2 food: enough to feed a specialist!
These are the Granary and the Waterwheel. These buildings can be seen as a growth helper, but it can be useful to look on them as specialist enablers -- building one allows you to assign a specialist without the city growing any more slowly than it would without assigning the specialist.
Of the two buildings, the Granary is the better deal, especially with the patch now given bonus food for wheat, bananas, and/or Deer, while a Waterwheel is something you might put in a dedicated GP farm, where you'll assign multiple specialists. Since you need to be by a river for both the Waterwheel and the Gardens (which gives a Great Person increase), this is a natural fit.
Later in the game, the Hospital gives +5 food, and later in the game is when you'll begin unlocking multiple specialist slots of a single type, so this building is a welcome late-game addition.
Specialists put Great Person points towards a great person of their type. Whenever that type's bar touches full, you get a person of that type, that bar is reset to zero, and the amount needed to fill all bars increases. This happily does away with 4 Great Artists spawning off their 1% chance à la Civ 4.
There are some social policies and wonders meant for specialists in the game, as well as one building:
Wonders
Statue of Liberty -- Replacable Parts -- +1 Production/Specialist
Hagia Sophia -- Theology -- +33% Great Person generation in all cities
National Epic -- Philosophy -- Requires Monuments in all cities (which you were probably going to get anyway) +25% Great Person generation in that city
Social Policies
Secularism -- Rationalism -- +2 Science to each specialist
Freedom -- Freedom -- Specialists are 1/2 as unhappy
Civil Society -- Freedom -- Specialists only consume 1 food
Democracy -- Freedom -- Great Person Growth increased by 50%
Meritocracy -- Liberty -- Free Great Person of your choice!
Building
Garden -- Theology -- +25% Great Person generation rate
Each Great Person has a special ability and also a terrain improvement they can be used for. All great people can start golden ages, amplifying production and gold income.
Different Specialists are used for different reasons and in different circumstances as follows:
Scientist:
Enabling Buildings:
University -- Education -- 2
Public School -- Scientific Theory -- 1
Research Lab -- Plastics -- 2
Scientists give +3 Research. Since base research is equal to population, Scientists are worth 3 population points in this regard -- without costing you the unhappiness of 3 population. Run scientists if you want to blow through the tech tree like nobody's business. This ability opens up mid-game with Universities.
Great Scientist:
Special Ability: Great Scientists can automatically discover a tech that is available to research. Research cost is irrelevant. Patching has made Great Scientists rarer; they're now available starting early-mid game, toning down some of the utility of the tech bulb. Nevertheless, always take this option with a Great Scientist; it simply surpasses the others.
Terrain Improvement: +6 research to the tile. The only time it's likely to be worth it to do this is if you have an extremely early Great Scientist (Meritocracy) and are planting them down in your National College city. Otherwise, researching a tech is going to yield more overall beakers, especially in the late game.
Artist:
Enabling Buildings:
Temple -- Philosophy -- 1
Museum -- Archaeology -- 2
Opera House -- Acoustics -- 1
Artists are almost the complete opposite of Scientists. They give +3 culture (HUGELY boosted by the current patch), which is used for expanding workable tiles and gaining social policies. Thus, artists find their best use in a loosely packed empire intending to grow each city to a large size, while Scientists find their best value in small, tightly packed cities. Unlike Science buildings, most culture buildings provide a flat culture rate rather than a percentage increase. Artists give you culture without the corresponding maintenance cost that culture buildings entail.
Great Artist:
Special Ability: Great Artists can perform a culture bomb -- if standing on a tile you own (or a neutral adjacent tile, according to posters below), they can convert every single tile around them to your side, unless the tile in question is an opposing city. This is the only way in the game to take a tile that another Civilization has claimed without war (though you'll take a diplo hit). The best targets for this are city-states, because they won't declare war on you (just get angry) and you won't have to keep shelling out gold to get their luxury resources, but if you want to seize control of a tile from a rival and aren't ready for a war, don't forget this option. There is a 10-turn minimum between Culture Bomb activation.
Terrain Improvement: +6 culture to the tile. This can actually be a worthy tile improvement if you're focusing on culture generation. This is the equivalent of 2 artists for less food investment, and the equivalent of three monuments without the maintenance or increased number of cities. It would be extremely good if there were more culture multipliers, but only the Hermitage and the Broadcast Tower multiply culture rather than providing base culture.
Merchant:
Enabling Buildings:
Market -- Currency -- 1
Bank -- Banking -- 1
Stock Exchange -- Electricity -- 2
Merchants saw a large boost in the last patch. The still generate 2 gold -- but that is no longer as much as a Trade Post, it's twice what a Trade Post (non-riverside) can bring in. Until you reach Economics, running a Merchant instead of working a Trading Post makes sense now.
Great Merchant:
Special Ability: Trade Missions provide a tidy bundle of gold and improved relations with the City-State the Great Merchant has visited. This suggests some synergy with Patronage if going that route.
Terrain Improvement: Customs House; this is essentially two Trade Posts on one tile. +4 gold. Since gold-enhancing buildings are all multipliers, this can actually be a good improvement -- if you will get more gold by working the Trade Post than you would have gotten from a Trade Mission. Both of these are much stronger now that the Trade Post tile improvement has been toned down, especially early on.
Engineer:
Enabling Buildings:
Workshop -- Metal Casting -- 1
Windmill -- Economics -- 1
Factory -- Steam Power -- 2
Engineers generate 2 Hammers, which is okay given that the patch has added more production in many tiles. Engineers are the rarest type of specialist in the game (in that their main enabling building, the Factory, requires a strategic resource to build), but the hammers can be run through some good multipliers.
Great Engineers:
Special Ability: Rush a Wonder. If you really need a wonder quickly, this will aid you, but otherwise, I'd actually give it a miss. It's a temporary production spike, but turn-to-turn production is far more versatile, which leads us to:
Terrain Improvement: +4 Hammers. Assuming a Grassland tile, that's 4 base hammers that feed themselves, which is overall the best production tile in the game. A Riverside Grassland gets even better with a Hydro Plant. If you're planning a super-production city, look to generate Great Engineers and plop them down as tile improvements. Even though production can be a bit more abundant with the new patch, several late-game buildings have actually had their costs increased, which means you'll want to make sure you've still got a production powerhouse at the end of the game.
As with Scientists and Merchants, though, the later in the game, the more likely the insta-build/tech/gold is to be the right choice, as you don't see as much return on the tile improvement investment.
Overall, with the new patch, specialists are evening out, though Great Scientists still win an award as the strongest Great Person type. You can even reasonably set up a GP Farm with the bonus food to the Granary. Specialize away!