It's just a painting like any other painting done on one or two pieces of glasses for foreground and background (traditional matte painting) or two layers (digital matte painting). It's used to create sets "studio setup" to save the money of building a new special studio for a battle scene for example.
It's used to create a cheap graphic effect like the one used in "lord of the rings" or "star war" and the technique is very easy to be applied.
In movies, the actor stands in front of a blank, usually green or blue, background after that the blank background is replaced with a background painting. A foreground painting with high details is put in the front so that it makes a realistic DOF (Depth Of Field) which helps making the whole scene looks more realistic.
A matte painting doesn't have to be painted. For example, when you watch a newscast, you find the host usually stands and there are some videos and/or an animated logo in the background which is also a matte painting technique.
Another very common example of matte painting can be found in photography studios where we pose, for example, in front of a garden painting (traditional matte painting) or a blank white, green or blue background replaced later with any other background(digital matte painting).