The Free Card
When you are in late position or last to act, you can raise with a drawing hand on the flop. This will likely make your opponents check to you on the turn, thus giving you the opportunity to check (if your hand does not improve) or bet (if you hit your draw). This will save you money if you do not improve and will reward you with profit if you hit. However, this move will backfire when you are re-raised on the flop. In these situations, it will cost you money but it remains a good play since you obtained information and have a good draw to a better hand.
When you posses a good hand and it is you turn to act, check in the hopes that an opponent will bet so that you can raise when your turn comes again. For example, you are in early position and have Ah-Qs. The flop is As-Qh-6c. You check and two players in middle position also check. A player in late position bets and you then raise. The reason for check-raising is to create a situation in which you can potentially hit a better hand, like a straight, but where it is too expensive for your opponents to call since, in this case, they do not have the correct pot odds with hands like gut-shot straight draws. If they still call, at least you have obtained information regarding the strength of their hands and forced them to pay as much as possible for trying to outdraw you.
Semi-bluffing is when you bet or raise with a hand that is not likely to be the best (at the moment) but you have many outs to outdraw your opponents if you get called or raised, although you are actually hoping to win the pot right there. For example, you are in late position holding Jh-Th and the flop shows Ks-6h-2h, thus giving you a flush draw with 9 outs. There are three other players in the pot and they all check to you. You bet without having the best hand but since they all checked, they indicated weakness and might fold pocket-pairs, a pair of 6's or 2's. Even if you do get called, you have 9 outs to the flush and maybe an additional 6 outs to win if you hit a J or a T, 15 outs in total. If called and it is checked to you on the turn, you have the option of taking a free card in case your hand did not improve.