The centrifugal compressor
The centrifugal machine works by flinging: air enters the spinning impeller at its eye and is thrown outward along the radial vanes, leaving the rim at high speed; the surrounding diffuser vanes then slow that flow, converting its velocity into pressure. One stage, one big pressure ratio, in a machine that is short, rugged and cheap. Its costs: a large frontal area, and poor stacking — feeding one impeller's rim flow into a second impeller's eye needs lossy ducting, so more than two stages are rarely worth it, and the advantages of adding a stage (more ratio) are bought with those duct losses, extra diameter and weight.
The axial compressor
The axial machine works by many small pushes: each stage is a row of spinning rotor blades followed by a row of stationary stator vanes. The rotor accelerates the air and adds swirl, raising its energy; the stator diffuses that flow, converting the added velocity into pressure and straightening the swirl to present the next rotor with clean axial flow. Across each stage the pressure and temperature rise a little and the axial velocity stays roughly constant; twenty such stages multiply into the huge overall ratios that make modern engines efficient.
Two design features follow from the physics:
Convergent air annulus — as pressure and density rise stage by stage, the same mass flow needs less cross-section, so the passage between casing and hub narrows toward the rear, keeping the axial velocity roughly constant instead of letting the densifying air crawl.
Twisted blades — a blade's tip travels far faster than its root (same rpm, bigger radius), so the blade is twisted along its length to present the correct angle of attack to the air at every radius; an untwisted blade would work well at one radius and badly everywhere else.
FOD and the two architectures
The two machines take a bird or a bolt very differently: the centrifugal compressor is far less sensitive to foreign object damage, its massive cast impeller shrugging off strikes that would wreck a more delicate machine, while the axial compressor is delicate: its long, thin, precisely twisted blades bend, nick and break, and one damaged early stage feeds disturbed air to every stage behind it. It is the axial machine's one great weakness against its otherwise winning design.
The diffuser
The diffuser is wherever the engine deliberately slows air to turn velocity into pressure: the vaned ring around the centrifugal impeller is one, and every engine carries another at the compressor's exit, a diverging duct that slows the flow before the combustion chamber, banking the last of the velocity as pressure and delivering air slow enough for a flame to live in — the problem the combustion chamber lesson opens with.