The Composite node allows you to use several images and output a single image. You can compare that to doing pre-compositions in editing software. You can connect several drawing, image generators and effect nodes in your Composite node and one bitmap (or vector in some cases) comes out of it.
Using Composites is necessary if you wish to apply effects to several layers combined. For example, if you want a character rig that is composed of several layers to appear transparent, you must connect all the layers of your character's rig to the same composite, and connect a Transparency effect to that composite.
NOTE Since composite nodes output a flat image by default, if the layers connected to it are on different positions on the z-Axis, the output will be a flat image on the topmost layer's position on the z-axis by default. For more information, see About 3D Input Output Options.