What is the difference between a two wire 220V hookup and a three wire 220V hookup as far as the neutral is concerned; i.e. how does the two wire work without a return? My well pump, air compressor and AC all use two wires with no separate neutral. My old compressor used 220V with each hot connected to each 110V leg, and a neutral connected to the load panel neutral buss. I thought the conduit would act as a return, but my well is connected to the house using PVC for both the water and wire enclosure, so there is no metal connection to the electrical system.

A circuit is usually takes 2 conductors to complete. With 220/240 it is between the 2 hots.
For 120, It is hot and neutral.
Recent? Code changes Made it necessary to have 4 wire dryers and 4 wire ranges.
In the past, ground was used to carry current, It was wrong.
Plus you could use 120 or 240 volt elements or parts.
When you say 220 volt, you say 2 conductor or 3 conductor, I think you mean, not counting ground.
Ground is necessary.
You Old compressor was possibly using white for ground.
Most 220 volt single phase, motors or compressors use 2 conductors to run and 1 ground. A 2 speed motor
Would have 3 wires, plus ground.

Take Care