The Venus flytrap, Dionaea muscipula, is probably the most famous of carnivorous plants, and for good reason. The spike-toothed traps that snap closed fast enough to capture fast-flying insects are unique among plants. Not only do they move quickly -- they're among the fastest things in the plant kingdom -- they do so only at a very, very precise trigger.
When an insect lands on the flytrap's trap (actually a modified leaf), it brushes against tiny hairs on the leaf surface. Touching those hairs triggers the trap, and it snaps shut. At first, there's still a little gap between the "teeth" of the trap, though, which lets very small prey escape -- that way, the plant doesn't have to waste its energy digesting something too small to be of any use. When too small a bug triggers the trap and then escapes, the trap can open back up within 12 hours. There's no need to waste time and digestive juices on a useless gnat.
If the trapped bug is big enough, though, it won't be able to slip out. Instead, it'll just keep squirming inside the trap: the continued pressure on the hairs tells the plant to seal the trap completely, pushing its edges together until it's a watertight pocket. Once that has happened, the plant releases digestive juices into the pocket. These juices break down and dissolve all the soft parts of the bug so that the plant can absorb the nutrients in it. Once this is finished, about 10 days later, the bug is nothing but an empty exoskeleton. The trap then opens back up, letting the bug-husk fall away and getting ready for its next meal.