STAGES:
single stage is what you have now, typically just one piston, only builds 125psi or so of pressure.
two stage puts the previously compressed air into another compression chamber, often after an intercooler, and then compresses it further up to 175 psi typically. This is better for industrial use or a larger shop.
I have never seen a 3-stage compressor that either of us could afford, so forget that!
PHASES:
single phase electric motor is what you have now, and what you want on your next compressor - that is what 99.9% of homes and most businesses are wired for.
large industrial environments are often wired 3-phase, which saves some electric cost, but I doubt you will ever need a 3-phase compressor, unless you inherit a factory. 3-phase is often found with 480V electric service.
there is no such thing as 2-phase electric any more (
except in Philly, but they are weird anyway - lol).
buying a 3-phase compressor on ebay or craigslist is often
not the great deal that it first seems, because you will have to either replace the large motor with a single phase version, or else use a phase converter, and neither of these two ideas are cheap!
AMPERAGE:
larger pumps need larger motors to power them, and this requires more AMPS (electrical current) - so if you need a lot of cubic feet per minute, you will need higher horsepower and hence, higher amperage
high HP/high AMP compressors are mostly needed in body shops or in shops where a number of techs are sharing the same shop compressor simultaneously.