A conservative force is a force which is not path-dependent. In other words, in moving an object from point A to point B, the total work done is independent of the path that the object took. The term conservative force comes from the fact that when a conservative force exists, it is possible to view the effects of the force in terms of a change in potential energy which keeps the mechanical energy conserved.
Nonconservative forces arise due to neglected degrees of freedom. For instance, friction may be treated without resorting to the use of nonconservative forces by treating heat as kinetic energy; however that means every molecule's motion must be considered rather than handling it through statistical methods. For macroscopic systems the nonconservative approximation is far easier to deal with than millions of degrees of freedom.