The definition of continuation bet can be summarize as a bet you make on the flop after you had already raised pre-flop. You're following up on your agressive pre-flop play by betting on the flop regardless of whether your hand improved or not.
There are many ways that playing aggressively pre-flop pays off.
EVERYONE FOLDS!
You win! End of story. Not much of a payoff, but at least you're stack is going up not down.
REDUCING THE FIELD
Okay, maybe not EVERYONE folds. And maybe seeing a caller or two causes your heart to sink. But the fact is that your pre-flop raise serves a purpose in reducing the field for the simple fact that there are fewer people with a chance to win. And if you let the guy with the suited 4-2 stick around to "take a shot," he's much more likely to catch his baby straight than if you had kicked him out of the hand earlier.
HITTING THE FLOP
Every once in a while, fate smiles and you actually catch a lucky card or two on the flop that improves you're already strong hand. Catching a set, pairing up on that Ace to go with your high kicker. These are the reasons to raise the action before the flop.
But sometimes things don't turn out the way they're supposed to, and you're left wondering why you would make such a silly play. This is where the Continuation Bet comes in handy. Simply put, the Continuation Bet is the bet you make after establishing yourself as the aggressor even if your hand doesn't improve. ESPECIALLY if your hand doesn't improve.
Whether you're first to act or in later position, continuing with your aggressive play shows your opponents a seriousness about the hand. It's up to them to decide if that seriousness is based on a strong pocket pair, two high cards that actually improved with the flop, or just someone making a continuation bet.
Make no mistake about it, a Continuation Bet is designed to win the pot right then and there. You don't really want action to go any further. And there are various circumstances to take into consideration before making the Continuation Bet. For example, a strong flop, say A-K-J renders your bet relatively useless. The fact that you had callers to your pre-flop raise should tell you that someone caught a sizable piece to this board. And a mid-pair wouldn't really bet into three over cards. In other words, a Continuation Bet in a situation like this would be seen exactly for what it is: a classic Continuation Bet.
All in all, the thing to hope for when your pre-flop raise is called is either a weak board or one where your hand markedly improves.