What is the difference between a single pole 120 V breaker and a double pole 240V breaker?

Answer for USA, Canada and countries running a 60 Hertz supply service.

A single pole breaker is used to supply 120 volts to a 120 volt load circuit and to protect the conductors of that circuit. This is one "hot" wire to the load and then back to the neutral. A two pole or double pole breaker is used to supply 240 volts to a 240 volt load and to protect the conductors of that circuit. This is two "hot" wires to the load without a neutral, eg. baseboard heaters, hot water tanks. The two pole breaker is also used with a neutral when the 240 volt device needs a 120 volt supply for its controls. eg. electric range, dryer.

If you mean a 120 breaker with two switches, those are designed to fit an extra circuit into a full breaker panel by removing the single, installing the double, and you now have two breakers in the space of one, each for the rated amps at 120 volts on the same "leg" of the 240 supply. These are not often recommended and some local codes do not permit them to be used more than temporarily.