There are some pro players - Fred Couples, for example - who do not wear gloves. They are very rare, however, and golf pros will always recommend the use of a glove.
The reason, as Johnny Miller has written, is that the human hand simply isn't as tacky as the material used in golf gloves. Especially when sweating or when grips are wet. A golf glove will provide a more secure grip.
Usually a golfer will wear one glove, on the lead hand (the lead hand is the top hand of your grip - for right-handed players, the left hand; for left-handed players, the right hand). Some golfers do wear gloves on both hands, but one glove is the norm.
If wearing a glove makes you uncomfortable, just be sure to use a towel before each shot to dry your palms, and always make sure your club grips are dry.