Passive DI boxes have the advantage of requiring no power. They use transformers to isolate and convert the output signal to a balanced line, and they are usually cheaper than active models. The downside is that they tend to exhibit a small amount of high frequency loss.
Active DIs are usually powered by a 9 volt battery, or by phantom power this is a 48 volt current normally supplied by your mixer - remember to press the +48V button though, or it wont work! Phantom powering is generally a better choice than batteries, especially for live use where a spent battery would spell disaster, but they do tend to be a little more expensive.