And we are talking skill, not athleticism. Skill, by definition, requires expertise.
Thus, while a sport like basketball is very demanding and generally requires a lot of talent, a person like Shaq, who couldn't shoot a free throw to save his life but was still a huge scorer and champion, is an example of why basketball is not on my list.
Some sports I've chosen are:
Golf - Some people do not appreciate how difficult golf is and how the smallest mistake could cost you a swing. Not only that, but golf is a sport that relies almost entirely on skill; an athletic person will not excel at golf just because they are athletic.
Gymnastics - I grew up watching a lot of gymnastics, my mom was a competitive gymnast and put my sister and I in gymnastics as well. My sister did well, and I dropped out at a young age because I was afraid to do a backflip. I excelled at many other sports, but gymnastics was not one of them. In fact, it scares me. Watching other gymnasts perform always amazes me. They also have different events (floor, beam, vault, etc). Take the beam, for example: people are doing twisting flips on a beam several feet in the air that is only FOUR inches wide. Skill is absolutely necessary and gymnasts spend so much time perfecting their craft. Very admirable.
Boxing - This is one of the most skillful sports there is, in my opinion. It requires so much, physically, but also requires brainpower. A simple error in judgment could lead to a person not only losing a match, but getting the shit beat out of them. They must be on their toes, literally. "Dancing" around the ring requires a great deal of skill in itself. (I'd also include other types of fighting such as MMA, but I don't want this to turn into a debate over which fighting style is more difficult).
Thoughts?

I would agree about golf, but I believe baseball is one of the most difficult of all sports, think of it this way if you can just get on base 30% of your at bats, that would make you pretty much a superstar. Boxing is a bit overrated even though I personally enjoy it, it's more about your genetics, and ability to endure a lot of punishment, but all professional sports require high levels of skill, and talent, along with training.