The Venus Flytrap (Dionaea muscipula) is a well-known carnivorous plant native to the sub-tropical wetlands of the US.
The 'trap' structure snaps shut when insects or spiders repeatedly touch specialised 'trigger hairs' on the trap's upper surface.
The mechanism which causes the trap to close in 1/10th of a second has been studied for many years - but has not yet been fully explained - see Wikipedia
The rapid trap closure of Dionaea muscinula Ellis has been explained by either a loss of turgor pressure of the upper epidermis, which should thus become flexible, or by a sudden acid-induced wall loosening of the motor cells. According to our experiments both explanations are doubtful."
Source : Planta, volume 179, pages 32–42
A 2005 study published in the journal Nature (archived here) suggested that the closure "results from a snap-buckling instability, the onset of which is controlled actively by the plant".
Although the molecular and cellular processes underlying the water movements that control anisotropic curvature changes remain poorly understood, we now argue that the macroscopic mechanism of closure is determined solely by leaf geometry."
Notes :
• The trap only closes if an insect touches one or more hairs more than once within a short time. A single touch does not trigger the mechanism. In addition, a trapped insect is also required to repeatedly touch the hairs after the trap has closed - otherwise digestive enzymes are not released, and the trap re-opens. Some researchers interpret thus as an ability to 'count' - thus implying 'cognitive' ability in the plant.
• A somewhat similar, and also poorly-understood trigger-trap mechanism exists in Dionaea muscipula the Waterwheel Plant. See : Planta , v. 174, pages 8–18.