There are several possible reasons for your betta's tail problem. Bettas are very susceptible to fin rot because they have such long, delicate finnage, so it's important to keep the water clean. He may also have torn his tail on your plant--is it plastic or silk? Plastic plants are notorious for injuring fins.
Fortunately, the problem doesn't sound serious. Just keep his water very clean (water changes twice a week), and also make sure it's around 80 degrees. A mini Hydor heater will do that nicely, if you don't already have one--bettas need them anyway, since they're tropical fish. You can also add a small pinch of pre-dissolved aquarium salt with every water change.
If you do all that, your betta should heal up fine