A non-conservative force is a force that turns some macroscopic motion (or mechanical energy) into microscopic motion (thermal energy). As long as all of the forces acting on a system (like a mass on a spring) do no net work, then mechanical energy (kinetic energy + potential energy) is conserved. As soon as non-conservative forces like friction or the normal force are introduced, the system gets warmer (increased thermal energy) and usually gives off sound (also a form of energy).
Non-conservative forces include:
All real systems have some non-conservative forces associated with them (the moon going around the Earth creates tidal forces, which warm the oceans slightly, but it's a small effect compared to the energy in the system). All systems lose some mechanical energy over time; this is part of the second law of thermodynamics. It's important to note that non-conservative forces don't destroy energy they just change it into a less useful (less ordered) form.
The University of Colorado has graciously allowed us to use the following PhET simulation. To get a physical intuition of how friction turns macroscopic motion into microscopic.
To learn more about conservative and non-conservative forces, please check out hyperphysics.