I'm not going to explain them really here (it'd take a lot of work to explain them), but at least cover some basics about some types.
Pre-BTS usual division was CE vs SE - Cottage Economy vs Specialist Economy. Later on those definitions have been further refined divided - I think Snaaty's use of Food Economy was one of the points where the definitions were reconsidered.
BTS introduced Espionage, and with it came Spy Economy or Espionage Economy that relies on stealing technologies.
I'll include BTS in consideration from now on.
Essentially any Economy explanation considers how does one get technologies. Production is sometimes considered (eg. it's noted that Food Economy can use whipping more easily than Cottage Economy for production of hammers), but more often than not it's assumed that you get production roughly the same way in all cases: specific production cities further dividing into military and wonder producers as deemed necessary.
Similarly Culture is not considered. Cultural victory examples most often imply heavy commerce (

) cities and slider to culture, with cultural and artist wonders, religions for cathedrals, and so on. This all goes outside the consideration of economy.
So, Economy generates Technologies. Gold is required to cover the bills (maintenance expenses from civics, cities, and military), with excess sometimes generated for upgrading or rushbuying when running Universal Suffrage. Gold does not give technologies, so it too falls outside the scope. Funny, isn't it?
For as long as we agree on focus of Technology generation, we can further divide Economy types fairly easily. What are the
direct sources of technologies?
1) Gained from other civs via diplomacy screens (traded, extorted, begged for, gifted)
2) Gained from other civs via espionage screens (stolen)
3) self-research with beakers
4) lightbulbed with Great People
5) gained from one-shot races like Oracle and Liberalism
6) popped from tribal huts
Huts and one-shots aren't viable for sustained economies, so we'll leave them out here. All other are possible to repeat.
Trading requires something to trade with. Most technology trading is done with technologies, so most trade economies generate technologies for trading purposes using other means. Trade technology that does not generate technologies by other means has to generate lots of gold and still has WFYABTA troubles - this has been successfully demonstrated by Kylaeran in an extreme variant. Still, as an extreme variant it does not fall within the scope of basic economies.
What we can conclude at this point is that any economy type must have sustainable source of technologies but may further enhance the economy by means of trading and by one shot deals like racing for Oracle or Liberalism, or by good luck with huts. Therefore, the basic division of Economy types here could be:
self-research with beakers,
lightbulbed with Great People, and
stolen from other civs. GP bulbing generally generates very valuable technologies and benefits greatly from trading. Self-reserch has to dedicate to beelines to get monopoly technologies that generate high value from trades. Stealing obviously will never generate monopoly tech, and trade values thus tend to be fairly low.
Self-research means beakers. Beakers can further be gained by very few methods: from commerce via slider, and from specialists. Generally specialists need representation to back them in beaker generation - running specialists without representation is generally done to generate great people for lightbulbing and trading with minor self-research only.
Commerce for slider can again be generated from working commerce heavy tiles, and from trade. Tiles will outproduce trade if wanted, but that comes at the cost of population working those tiles.
So, there are some major economies with supporting economies. Major ones are:
1) Cottage Economy
Heavy cottaging leads to high commerce, but population is tied to working those cottages. Generally city growth is slower than in other economies, and whipping is not as beneficial. Cottage economies start slow but in late game they're often the best - Printing Press and Free Speech are major boons, with Universal Suffrage providing additional production, Emancipation boosting growth of newer cottages.
Great people are generated by specialized GP Farm city that has lots of foods and only runs specialists - lots of them.
Production is by specialized production cities, eg. Heroic Epic city for military, possible Military Academy city for secondary military, and Iron Works city for heavy late wonder production for example.
Gold is generated often by running slider at below 100% - it's often assumed that slider at 60% or higher represents reasonable minimum for stable CE. Gold can be generated by having dedicated Merchant city (essentially GP Farm that runs merchant specialists), running a merchant or two in many of the minor cities, or by the luck of having Shrine of major religion.
2) Representation Specialist Economy
The first thing to do: build the Pyramids. That gives access to representation civic early. This way scientist specialists generate six beakers each. Also, by running specialists in many cities great people will be born more often. GP's provide secondary economy, but may as well be used only for bulbing key technologies (most common is Philosophy, Paper, Education in Liberalism race), rest used for Academies, settled, or other abilities.
As said, Great People are generated naturally as byproduct of research. Additional GP's can be generated by specialized GP Farm, but this is not necessary.
Production as with CE, except any city that has hammer-tiles can become temporarily a production city by whipping and moving specialists to working mines.
Gold is usually generated by keeping the science slider low - even at 0% - with commerce thus becoming gold. Commerce comes in small amounts from many sources, but main source is usually one or two cottaged cities - often capitol is one for Bureaucracy civic boost. Trade economy support can also be used to generate significant commerce for gold. Shrines and Merchant specialists are possible as with CE.
3) Great People Lightbulb Economy
GPLE is a subset of Specialist Economies in that specialists are in key role. Here Pyramids (and thus representation civic) are not assumed, and thus actual beaker production of scientist specialists is not enough to make up a full economy. Beakers are rather commited to cheaper old "backfill" techs as well as prerequisites that allow Great People to lightbulb specific important technologies. These lightbulbed techs are further traded for many others, as they are assumed to be highly valued monopoly techs. Techs gained from trade can further be traded for even more (assuming no No Tech Brokering option), leading often to GP providing 3-5 times the beakers of the technology bulbed directly.
Otherwise as Representation-SE.
4) Espionage Economy
Main point is generation of Espionage Points against other civs to allow for stealing of technologies. This can be achieved by running slider high on Espionage, but in itself that is fairly inefficient. To make EP generation more efficient, you want to generate Great Spies to build Scotland Yard (+100% espionage) in high commerce city, settle in Scotland Yard city, and possibly to infiltrate target civs for instant beaker gain (similar to lighbulbing).
Further you need Alphabet tech to train spies who will then go do the dirty work.
Overall there are many similarities to CE, with focus on EP's instead of beakers.
There are many things one could write about regarding these economies, including introducing other economies to complement them. But I'm not doing that today (if ever), so I hope you can find more information from other sources.