Game engines are the core software to a game, to put it simply, without them there would be no game. The software will crunch the code written by programmers, and construct the virtual world that you are playing in and everything in it.
There are many different types of engines, some created for specific tasks. For example the widely used Havok engine, used to simulate realistic physics and 'Speedtree', which was used to generate the realistic trees and vegetation in The Elder Scrolls IV : Oblivion. These types of engines are know as middleware and are usually deigned be used in conjunction with other engines.
Many game developers will buy into an existing game engine to use on their game, which will save on time developing a new one and cut down on man power. A different approach will be to develop a new game engine or improve an existing one. This would enable developers to create the engine to suit the experience they want to make.
A good example of engines being used together in a next-gen game is in Star Wars: The Force Unleashed. This game uses the Euphoria engine, used to generate realistic human responses with other humans and the game world, with the Havok physics engine and the DMM Engine, which regulates what objects are made from what substances and how they shatter and break accordingly. The game appears to push the boundaries of physics in gaming, and you get to be Darth Vaders apprentice, how cool is that!?