A Hohmann Transfer is an orbital maneuver that is used to transfer a satellite or spacecraft from one circular orbit to another. It was invented by a German scientist, Walter Hohmann in 1925 and is often (not always) the most fuel efficient way to get from one circular orbit to another circular orbit. Although it can take longer time as compared to other orbital maneuvers.
To understand this transfer phenomena, you need to understand couple of things first.
First, each orbit around a planet as got a unique orbital energy and a unique velocity associated with it. A circular orbit of radius 1000 km, an elliptical orbit of apogee 1000 km and perigee of 600 km, a circular orbit of radius 1200 km, all will have different orbital energies.
Second, Hohmann transfer works only when both the — initial and final orbit, are nearly coplanar, i.e. in the same plane.
Third, the difference between the energy levels of two orbit is uniquely co-related with Delta-v, that is a measure of the impulse per unit of spacecraft mass that is needed to perform a maneuver from one orbit to another.
A Hohmann Transfer is half of an elliptical orbit (2) that touches the circular orbit the spacecraft is currently on (1) and the circular orbit the spacecraft will end up on (3). It takes two accelerations to get the original orbit to the destination orbit. To move from a smaller circular orbit to a larger one the spacecraft will need to speed up to get onto the elliptical orbit at the perigee and speed up again at the apogee to get onto the new circular orbit. To move from a larger circular orbit to a smaller one, the processes are reversed.
The transfer (yellow and labeled 2 on diagram) is initiated by firing the spacecraft’s engine to accelerate it so that it will follow the elliptical orbit. This adds energy to the spacecraft’s orbit. When the spacecraft has reached its destination orbit, its orbital speed (and hence its orbital energy) must be increased again to change the elliptic orbit to the larger circular one.
Due to the reversibility of orbits, Hohmann transfer orbits also work to bring a spacecraft from a higher orbit into a lower one; in this case, the spacecraft’s engine is fired in the opposite direction to its current path, slowing the spacecraft and causing it to drop into the lower-energy elliptical transfer orbit. The engine is then fired again at the lower distance to slow the spacecraft into the lower circular orbit.