A continuous stream of air flows through the air-breathing engine. The air is compressed, mixed with fuel, ignited and expelled as the exhaust gas. The air-breathing engines are lighter than rockets, because they utilize ambient air as oxidizer. On the other hand, their maximum velocity is limited to 1-3 km/s due to extreme temperature and dissociation of the exhaust gas.
Turboengines designed for the maximum velocity of less than 1 km/s are light-weight, durable, and have a high specific impulse.
Ramjets can operate at a higher velocity than turboengines because a smaller area of the engine is exposed to the extreme heat. Their specific impulse is lower and varies dramatically with velocity.
The air-breathing engines can be used instead of the expensive first stage of the chemical rocket launcher. The idea is called airplane-rocket relay.
The maximum velocity of a hydrogen-breathing engine is about 4 times higher than the maximum velocity of an air-breathing engine of the same design. It is possible to fly the engine in a long tunnel cut in antarctic ice sheet and filled with hydrogen (ice gun). Another option is to fly the engine in a disposable tinfoil balloon filled with very cold hydrogen gas (ramjet in balloon).
AIAA Air Breathing Propulsion Technical Committee.