Couplings are used to "couple" a pump and motor or engine together. One coupling half slips over the shaft of your engine or motor, one coupling half 

slips over the shaft of your pump, and the flexible insert goes in between to mate the two halves together. The higher the series of your coupling, the more robust the coupling.  The close machining tolerances of these couplings assures vibration-free operation and easy, accurate alignment without need for special tools.