If the work done is path independent, the underlying force is conservative.

Example: the force of gravity:    dW_grav = mg • dr = - mg dy
W_grav = - mg(y_f-y_i) = - mg Δ y

Because the work done depends only on the endpoints and not on the path the underlying external force is conservative.

Examples of conservative force:   gravity, springs.
Example of non-conservative force:   friction.

For an arbitrary conservative force:   W_cons = - (U_f - U_i)

Energy Conservation Statement:

Start with our previous connection between kinetic energy and work:

W = Δ K   →   W_cons + W_non-cons = Δ K

U_i - U_f + W_nc           = K_f - K_i

K_i + U_i + W_nc = K_f + U_f