A conservative force is a force that conserves mechanical energy. This means that all of the macroscopic motion (or mechanical energy) stays as macroscopic motion and doesn't turn into microscopic motion (thermal energy). As long as all of the forces acting on a system (like the Moon going around the Earth) are conservative, mechanical energy (kinetic energy + potential energy) is conserved. That means that the system doesn't get warmer (increased thermal energy). The energy can still change forms for example the kinetic energy becomes potential energy and then back again.
Gravity is an example of a conservative force. A child turns the same amount of gravitational potential energy into kinetic energy from going down a spiral slide, a straight slide, or jumping off the slide. Friction is a non-conservative force—the friction down the slide does negative work on the child, slowing her down, and no work if she jumps (which is why it's safer to slide down a slide then jump off the top).
To learn more about conservative forces, please check out hyperphysics.
This simulation was provided by the University of Colorado, play with it to see how gravitational potential energy and spring potential energy go back and forth and create a changing amount of kinetic energy (hint: click show energy before hanging a mass):