Trying to offer the best combination
The City Governor (City Automation) will try to offer the best combination of citizens working tiles and citizens becoming specialists without sending the city into starvation. It's not uncommon for cities with a reasonable food surplus that the City Governer may opt to start running a few specialist citizens if it's able to (e.g. if the city has a Forge and a good food surplus, it's quite likely that the City Governor will run a Specialist Engineer).
I suppose it's worth saying that the more you use the computer's choices (e.g. City Governors, Automated Workers, Automated Exploring) the less likely that you will get 'an edge' over the A.I.. I am not necessarily saying don't use City Governors, but I would be mindful that the computer doesn't understand your strategy as well as you do.
You can emphasise Great People
You can use the 'Emphasise Great People' selection, which is the

button while the City Governor is on, which if possible will see citizens turned into specialists. Obviously the more specialists you run, the faster your yellow

bar will fill, and you'll pop a new Great Engineer, Great Artist, Great Merchant, Great Scientist, or Great Prophet.
Manually placed specialists and the City Governor
You can override the City Governor's settings without turning the City Governor off by pressing the plus "+" symbol next to the Specialist that you want. A thick yellow square around the specialist symbol will indicate that this specialist citizen is 'forced' by the player rather than necessarily being the City Governor's selection. The City Governor can still be used for placing other citizens while having forced specialists.
Choices can be influenced
Some conditions will affect the attractiveness of specialists - for instance the Angkor Wat wonder provides Specialist Priests with an extra

, so the City Governor may look to running more Specialist Priests than usual given the right conditions (food surplus, buildings that allow Specialist Priests to be run and/or the Caste System).
You can also use the 'emphasise' buttons in combination - e.g. 'emphasise

' and 'emphasise

' in combination will probably see the food surplus drop to +1

per turn, and the city run the maximum possible Specialist Scientists, and if there's no more Specialist Scientist spots available, it may look at running Specialist Artists because they also contribute

.
I hope that this brief overview is of some use. I've opted not to go into the merits and pitfalls on assigning specialists, or all of the influencing factors, or the purity of Great Person pools, or how to set up a Great Person Farm, etc. etc., but there are lots of discussion threads on these topics.
(Edit: Cross-post with Anita)