First of all they will do whatever is needed to fulfill their needs, then the main things that sims do are: play on the computer (if they have one), prepare food (even if they're not hungry) and interact with other sims on the lot; those are the ones they do most.
But they can also read a book (if they have bookworm trait they will do this more often), play an instrument (if they have the skill), kleptomaniacs will steal stuff, kids and teens do their homework (they will do it on their first chance if they have some nerdy trait, but if they are rebel they will never do it on their own), adults will go to work on time (though, they will not look for one, you have to choose it), they may ask other sims to marry if their relationship is good enough and they like him/her too... stuff like that.
You may want to take a look at this page: http://sims.wikia.com/wiki/Free_will
Free will does not only keep a Sim sustained in life with almost no player interaction, but will also choose interactions based on the Sim's traits, favorites, and how they are compatible with other Sims in socialization. For example, if a Sim has the Green Thumb trait, they will autonomously garden and take care of their plants and even talk to them. If a Sim knows the recipe for their favorite food they will be more likely to cook it than any other dish. Sims will give positive reactions and interactions to Sims who are compatible in traits and the opposite with non-compatible traits. With story progression on, this free will is more noticeable with other Sims not currently controlled by the player. Sim children will even do their homework if the free will is high enough. When free will is set to off, then Sims react autonomously only in order to save their lives.