If your betta is housed in an uncycled bowl or tank then you need to do regular water changes to remove the toxic ammonia that builds up in the water, as it will quickly make your fish sick or even kill him. But how do you know how often you need to do this? All fish produce ammonia, but all fish produce it at different rates. The only way to really know how often you need to change the water is to use an ammonia drop test kit, such as this one: http://www.petsmart.com/product/index.jsp?productId=2754002
Here's what you do: Fill the bowl/tank with fresh, dechlorinated water. Each day test the water for ammonia. Eventually, depending on the size of the container, you'll get an ammonia reading. So you'll need to start doing water changes on the previous day. For example, if you get an ammonia reading on day 5, then that means you need to change 100% of the water on day 4. The goal is to never have enough ammonia in the tank that it is readable on a test.
Repeat this once or twice the first few weeks you have the fish to make sure that you have a good schedule down, and then you really won't have to worry about it too much anymore.
A few tips:
Don't overfeed! Less food = less poop = less ammonia = less water changes. Your betta won't starve on just a pellet or 2 each day or even every other day!
Remove any uneaten food from the tank as it will rot and add to the ammonia.
Buy a turkey baster from the grocery store (do NOT use the one that's used on the thanksgiving turkey!) and use it to remove uneaten food and poop from the bowl/tank each day.
Be sure when doing water changes that the new water is the same temp as the old water (buy a thermometer from the pet store - they are cheap!) and that you have used a proper dechlorinator in the new water.
It's much easier to do water changes in a bowl/tank without gravel. If you like the look of having something on the bottom of the tank then a few large, smooth pebbles or marbles work well.
Hope this helps!