A single-stage furnace has only one stage of operation, it is either on or off. This means that it is relatively noisy, always running at the highest speed, and always pumping out the hottest air at the highest velocity.
One of the benefits to a single-stage furnace is typically the cost for installation. Single-stage furnaces are relatively inexpensive since the technology is rather simple.
A two-stage furnace has two stages of operation, full speed and half (or reduced) speed. Depending on the demanded heat, they can run at a lower speed most of the time. They can be quieter, move the air at less velocity, and will better keep the desired temperature in the house.
A modulating furnace can modulate the heat output and air velocity nearly continuously, depending on the demanded heat and outside temperature. This means that it only works as much as necessary and therefore saves energy.