I don't own one but have known several people with them and had a close friend work at a primate rescue sanctuary which I was able to visit on several occasions.
Primates make horrible pets. There's really no other way to put it. They are highly social animals that are extremely intelligent, extremely active, and being outside of a natural group, you the owner have to fulfil all of their requirements including the social ones without messing them up psychologically. They require almost constant interaction, much more than captive parrots, which I would consider one of the most demanding pets as far as daily interaction goes.
Unlike a pet such as a dog, training primates is difficult. They view a person as a conspecific and not a master. This introduces a number of problems in the owner pet relationship, simply because they don't consider themselves a pet. A similar mentality is an animal like a raccoon, which also makes a horrible pet, but similarly as when you try to get one to do something that it doesn't want to do, it's simply not going to happen.
More importantly, meeting the social requirements of a primate is virtually impossible. Primates have complex societal structures and relationship hierarchies. Since a single human or even a family cannot provide a similar social structure, what most people end up with is a very capable, strong, and agile, but mentally handicapped animal, that's going to live from 25 - 50 years. Like humans these animals have emotional states that can be completely overwhelming. They can also suffer from psychological illnesses such as depression and anxiety. These do not usually manifest pleasantly. As stated in another answer it's like a perpetual toddler, except it's a really fast toddler that can jump, throw things (including bodily fluids and excrement), bite and spit, and scream, but with many degrees more intensity than a human all while having the mental stability of a teenager. If you see how the interactions between same species monkeys is not always pleasant, just remember that you are the other monkey when you own one. Something that many don't consider is also that they can share many diseases with humans and finding suitable veterinary care isn't always possible.
As far as selection goes, larger primates are dangerous and completely unsuitable as pets. Chimpanzees especially are very aggressive once mature and many times stronger than even an abnormally fit human. For that matter all apes should be completely outside of consideration as well as large monkeys and anything endangered, which includes a large percentage of all primate species.
If you're still legitimately interested in keeping one as a pet, I would strongly advise to first find a sanctuary or institution where you can get more information and some hands on experience with them. By the time you spend a few months with them, you should have a pretty good idea if you actually want anything to do with keeping one in your house, I suspect you will not. But, you'll have enough of a foundation to make a legitimately informed decision on keeping one.