Game engines are essentally what run the game and tie all the elements of programming , art, animation and sound together. There are a whole variety of different game engines around that are used for different reasons as they all have their ups and downs. A company always has a few options in how they get their game to run. They can use a pre-made game engine such as Unreal or they make their own in house engine, this of course is down to the amount of money and time the company making the game has and can afford to spend.
There are other options like using middleware programmes which are basically smaller game engines that help create certain different effects within the main game engines react. For example the DMM engine allows objects in game to have assigned material properties and effects they way objects in game respond to different effects on it. In other words wood will splinter, snap and burn but glass with shatter crack and melt. This software was used in the Force Unleashed and allowed the characters and world to respond in a more realistic way with each other.
Game engines have also in recent years become more user friendly so that it doesn't take a team of programmers to build a level and import objects in. Now the artists can build the level and assests realitively easily and are able to render and light their scene and have more control of where they want everything to go.
There is also a major difference in the way game engines can build levels, they are either additive or subtractive. Its pretty straight forward in the differences, a subtractive game engine means that you start off with infinite solid space and you carve into it and create the world in which you move in. An additive (also named Relative) engine allows you to build and create your world and environment in an empty space usually called a void by creating the solid forms and working out form there. The Unreal Engine has the ability to allow users to do both which makes it a very versatile game engine to use, opening itself up for wider uses.
The introduction of game engines has allowed games to be made more efficiantly and cheaply, without them many games wouldn't be able to have all the great visual aspects that make them so great.