An active defense is a defense which takes an active role in defending a weakness. This is in contrast to a "solid" move or a passive defense, for which the stones will have a purely defensive role. Example:
Black 2 can be seen as a "solid" move, or a passive defense. Imagine choosing instead a pincer, let's see what might happen:
Note that black 4 has "by the way" helped to protect black's bottom. It is still not secure yet, but black has made a move in this area; therefore we can say he has employed an "active defense" of this area with the pincer at 2. Also, black 2 has a good chance of living. This is a well known joseki so black isn't going to get more than white; but there may be some situations in middlegame fighting where by employing an active defense, you can come out ahead. Sometimes, the solid move is the best move; in other situations it's slack and an active defense is necessary. In yet others the active defense move is an overplay, so you must be careful to balance the slack with the overplay.
Active defense is preferred by people who play in the Taiji Style. It's a necessary component of the style. To properly perform an active defense (or active attack!) you should be well grounded in strategic concepts such as motare and leaning attack. kikashi (forcing move) is also a key concept. Learning to recognize kikashi wherever it appears can make you a very strong player.
What are the follow-ups? Black 2 seems quite isolated and can be subject to a pincer.
Tapir: A typical continuation is a.