Definition of continuation bet
A continuation bet is a bet that a player makes when he has the initiative. To have the initiative, you must have made the most aggressive play on the previous street. For example, a player limps before the flop, I isolate him by raising to 5 big blinds and he calls. I have the initiative. If I decide to bet on the flop, I'm said to have made a continuation bet. If my opponent calls and I bet on the turn, I am said to have made a continuation bet on the turn. A continuation bet on the turn necessarily implies that a continuation bet was made on the flop. If I don't bet the flop and I bet the turn and have the initiative, then I am said to have made a delayed continuation bet.
History of the term
Dan Harrington is credited with coining the term and theorizing it in his trilogy on tournaments. The term is now widely used. Trackers like Poker Tracker 4 and Xeester also use it in their statistics. A continuation bet is not a standard bet. If you analyze your opponents, you will notice that they have a higher continuation bet frequency than their general betting frequency.