A 220 volt breaker is two breakers tied together with one tie handle. It bridges across both bus bars to allow 220 volts to pass through it. A twin breaker is two breakers that fits in the sapce of a single breaker. There is 2 handles for two independent operated 110 volt breakers. These breakers are called, twins, half sized, or mini breakers. Sometimes they are used a cheater breakers. There is an exception where you can tie two twins together to get 220 volts. When you install a twin breaker in a 20/20 panel, you can get 40 circuits in a 20 circuit panel. That is not allowed as per the manufactures instructions or as per sec. 110-3(b) of the NEC.
A "twin" breaker provides two (2) circuits from one (1) buss space; both at 120 volts to neutral. There is one (1) physical connection to the panel buss. They are also refered to as "piggybacks" and "tandem"
A common use of the above is to "add" additional circuit capacity to a "full" panel. The design intent was/is to provide a greater circuit capacity in a physically "smaller" cb panel. (See yoyr 20/20-24/4- post)
A "220" breaker, as you have referred to it, is a "two pole" breaker. It is basically two single pole, full size breakers with a common handle & trip. This breaker connects to two buss spaces, and provides 220 volts (or 240, or 208).
There are also some "twin" 2 pole breakers around, but that will really add to your confusion.