It depends on the type of system, but essentially you can think of it as a way something can move. If you think of a mechanical movement, there are 6 basic components that something can move in 3D space. It can move linearly in the X, Y and Z directions. It can also rotate about the same three axes. Gneral motion is a combination of those 6 movements. It is said that the body has 6 degrees of freedom if it is not constrained in any way.