What caliber handgun is the best for self-defense?

Shot placement is FAR more important than caliber. A hit with a .22 is more effective than a miss with a .45 ACP or a .44 magnum. So really the best caliber is the one you are most accurate with. However, you may not have a lot of experience with various calibers so here are some notes regarding the "popular" self-defense calibers:

1. 9mm, effective if using jacketed hollowpoint ammo. Ammo is cheap and available worldwide.
2. .40 S&W, proven street cred and used by most police in the US. Not really used outside the US. Ammo can get pricey.
3. .45 ACP, a slow heavy bullet that will knock someone down, but these handguns are not easily concealable without a jacket or heavy sweatshirt. Ammo can get expensive.
4. .38 Special, primary police round until the 1980s, effective but considered inferior to the 9mm, .40 S&W, .45 ACP, etc.
5. .357 magnum, probably one of the best rounds out there...but not good for self defense in a small area due to the loudness of the blast and the fire that will shoot out of the barrel (could cause temp blindness at night). However, the best self-defense round in a wilderness setting or if you have a lot of land.
6. .44 magnum, very effective round but many will be unable to handle its recoil, experienced shooters only (especially if you want to be accurate).
7. .380 Auto, considered the minimum round for effective self-defense. Once again shot placement matters more than the round, but I would pass on this round and opt for a 9mm (9mm has performed better in tests and the ammo is cheaper too).
8. 357 SIG, very fast, higly penetrable round. Effective but is prone to overpenetrate...this is bad if you live in condos or apartment buildings. Also, probably not the best round for an urban self-defense scenario given its penetration ability. Ammo is really expensive, considered a niche round.

That covers most of the "popular" rounds. Below is a link to an article regarding handguns and self-defense:

http://www.keepandbeararms.com/information/XcIBViewItem.asp?ID=3195

As for my recommendations:

9mm, .40 S&W are good choices for semi-autos.
.357 Magnum handgun is the best choice for a revolver, because it can also fire .38 Specials. But a .38 Special handgun cannot fire .357 magnums.

Good luck and make sure you take some type of firearms class or course if you are unfamiliar with firearms and there use.