Mathematically, the degrees of freedom of a system is the number of independent parameters that define its configuration or state.
In Mechanics, a train on a fixed track has one degree of freedom because its position can be defined solely by the distance along the track. A car has two degrees of freedom, because it can move freely along the surface of the earth. It can depart from a north-south road and go east or west, for example. An airplane has three degrees of freedom. It can move like the other two but now vertically as well.
In order to describe the motion of the airplane, you would need a Real Vector Space of order three. The order means the number of degrees of freedom – or dimensions – of the space.
In other fields, other parameters may be used. For example, heat, charge, etc.
In thermodynamics, the degrees of freedom refers to the number of independent variables or parameters of the thermodynamic system.