The return spring is the spring that brings the handle, the knob or the latch back into place after use. It is what makes the handle return to horizontal when you let go and the latch spring out by itself to catch in the strike plate.
Depending on the model, it can sit inside the lock case or in the handle itself, in its rose or in its backplate.
When the spring wears or breaks, the handle droops and the latch does not come all the way out, so the door does not catch when you close it. On the coast, damp and salt air make it rust sooner. It can usually be replaced without changing the whole lock. More under lock installation and lock repair.