stage 1 - 2 - 3 are just terms given to different types of map really by tunners,
But basically:
Stage 1 - a generic map that is written to be applied to a 100% standard car, typically these will take your 150bhp A3 to 200bhp/220lbft , or your 210/225 S3 up to around 245-250bhp/ 260lbft
A stage 2 map is a custom(ish) map, whereby ususally a base map is written to suit a certain group of mods, normally things like a turbo back exhaust, induction kit, and a front mount intercooler, things like custom code stage 2 for the AGU is just a generic map still, but written to suit the AGU with bolt on mods. Some tuners will use a pre written base map for a stage 2, then alter it slightly to suit the specific car
this will typically take your A3 to around 230bhp/250lbft, or an S3 to maybe 260-265bhp/280-300lbft
Stage 3 is a totally custom map, written to suit big turbo aplications/hybrids with larger injectors, possibly a change in maf housing (and therefore scaling), and pretty much anything else that's unusual and not usually done. Power output from this is entirely depending on the spec of the engine/turbo, although of course, you wouldn't be doing it unless you were going for more than stage 2 power....