A centrifugal pump consists of an impeller rotating within a casing. Liquid directed into the center of the rotating impeller is picked up by the impeller vanes and accelerated to a high velocity by the rotation of the impeller and discharged by centrifugal force into the casing and out the discharge. Thus the name centrifugal pump. (Reference Cameron Hydraulic Data pg 1-4)
A vortex pump is centrifugal. It works on the principle that the impeller will swirl the liquid. These are used typically with a stringy or fibrous material to pass though the pumps without binding or clogging. It may also be able to handle larger solids. The efficiency of the vortex or recessed style pump is normally much lower than an enclosed impeller. For example 40% vs 70% efficient.
Another type pump is reciprocating positive displacement pumps. A simple explanation is a piston moving back and forth with inlet check valves letting liquid in on one stroke and the liquid traveling out a discharge check valve on the other direction of the stroke. Warren Rupp makes an air powered diaphragm pump which is an example of a recip pump.