"Conservative" forces conserve the total energy of an object and non-conservative forces don't! That, at any rate, is where the name comes from. Since total energy of an object is the sum of kinetic energy (depending only on speed) and potential energy (depending only on position), if you move an object around with only conservative forces involved, finally returning it to its orignal position and original speed, you have not changed the total energy and so have done no net work. Gravity is an example of a conservative force. The force moving planets around the sun returns then, eventually, to the same point in their orbit with the same speed and so does no net work. That's why gravity doesn't "run out"!
Friction, on the other hand, is a non-conservative force. If you move a refrigerator across the kitchen you are not changing its height and so not its potential energy. If after moving it across the room and leaving it standing still you have not changed its total energy. But you certainly will have to do work! You have to do work to overcome friction- that work goes not into the energy of the refrigerator but causes the floor to be slightly warmer.
In addition, the potential energy can only be defined for the (minus) work of a conservative force. The key is that the work depends ONLY on the starting and ending points, and not on the trajectory. Otherwise (for a non-conservative force) the path has to be specified, so it makes no sense to speak of a potential energy which should depend only on the position.
The path-independence of the work is equivalent to saying that the net work (due to that conservative force) in any closed path is zero.
I guess the sign is conventional (although i don't think anyone's going to change it now) and also you can add an arbitrary constant to the potential energy and get the same physical results, since the quantities of interest are energy differences.