The term rotary pump is usually given to a positive displacement style of pump that operates in a rotational manner rather than a reciprocating manner.

The multi-stage pump is usually a centrifugal pump with more than one impeller. They are quite different.

In most cases, the nature of the application will lead you towards the type of pump required. Applications involving viscous liquids generally use a positive displacement pump of some type. Higher pressure and clean liquid applications will frequently use a multi-stage pump. Most others will use a single impeller centrifugal pump.