It depends on the type of pump and the situation. Centrifugal pumps cannot prime themselves because they rely on some clearance between the impeller and the pump casing. There is simply not a sufficient seal to draw the weight of water into the eye of the impeller.
This doesn't mean you cant have a self priming centrifugal pump though. Some will use a vacuum pump on the suction side to evacuate the air. I've also used one which incorporated a peristaltic pump to do the same.
Very crudely I've also fitted a NRV to the end of the suction pipe and filled the pump through the discharge.
If you run a centrifugal pump dry you can total ruin it. Most rely on the flow of water through them to cool the seals. Having said that I've seen one run dry (by people who thought they where doing the right thing???) and the paint on the outside of the pump casing was blistering because it was so hot. I replaced the mechanical seal and packed out the impeller to get the correct clearance from the wear plate and the pump was fine.
So for all that I would say that we prime pumps to get them to work. Simple