The bridge is a piece, usually wood on acoustic instruments, on which strings rest and are sometimes attached. The vibrations of the strings are transferred to the resonating body of the instrument.
A saddle is a bone, ivory, metal, wood, or synthetic piece which is set atop a bridge, usually in a slot or groove. The strings sit on top of it. Traditionally saddles are not glued to the bridge.
Sometimes a second piece of material is glued to the top of a bridge. If it is fitted into a notch or slot, it is called an insert. If it is the top layer of a bridge, it is called a cap.