What do compressor/turbine stages mean, and why we need a different number of stages?
For axial compressors (what you'd find in a typical turbofan), each stage is:


A rotating disk of blades
Stationary blades that follow.


They're called rotors and stators, respectively. So those two rows of blades are one stage.

The rotors increase the air's velocity, the stators turn this velocity into pressure, and repeat (see image below). The following stage would compress the air more, and so on, incrementally increasing the pressure.

Likewise but in reverse for the turbine stages (the fixed blades also come first).


Source: The Jet Engine (book)