Do plant eat insects?

A carnivorous plant (sometimes called an insectivorous plant) is a plant that derives some or most of its nutrients (but not energy) from trapping and consuming animals or protozoans, especially insects and other arthropods. Carnivorous plants usually grow in places where the soil is thin or poor in nutrients, especially nitrogen, such as acidic bogs (moss in Scotland) and rock outcroppings. Charles Darwin wrote the first well-known treatise on carnivorous plants in 1875. There are five basic trapping mechanisms that are found in carnivorous plants. These are:Pitfall traps (pitcher plants), which trap prey in a rolled leaf that contains a pool of digestive enzymes and/or bacteria; Flypaper traps, which trap prey using a sticky mucilage; Snap traps, which trap prey with rapid leaf movements; Bladder traps, which suck in prey with a bladder that generates an internal vacuum; Lobster-pot traps, which use inward pointing hairs to force prey to move towards a digestive organ. These traps may also be classified as active or passive, depending on whether movement aid the capture of prey. For example, there are both passive flypapers, such as Triphyophyllum, which secrete mucilage, but whose leaves do not grow or move in response to prey capture; and there are also active flypapers, such as sundews, whose leaves undergo rapid growth, aiding in the retention and digestion of prey.

The one that I know of is the venus fly trap. It looks like an open clamshell and once the insect is inside the sides close up on it.

Yes there are, you can check the biology books for insectivorous plants topic.