A classical rocket has one, two, three, four etc. stages to make orbit, each smaller than the previous, each burning for a time, before the next one lights.

However most rockets, such as The Space Shuttle, Ariane, Atlas have more than one set of engines running in parallel at lift off. At some altitude one set (the half stage) of engines (and often associated fuel tanks) are dropped off, and the remaining stage continues burning all the way to orbit. This is known as One and a Half Stages to Orbit.

The biggest advantage of this idea is that total size and weight of the engines can be smaller and lighter, as the 'half stage' engine gives lots of extra thrust early on, which means that more fuel can be lifted, initially, and more payload.

In fact the half stage's engines give more thrust than the main ("sustainer") engines- so they quite literally lob the rocket up. At separation the remaining stage actually doesn't have enough thrust to lift the rocket on its own, not until it has burnt off most of the fuel; so the sustainer engines are then in a race to burn off enough fuel so that the acceleration on the lighter rocket is sufficient to make orbital velocity before falling back down to earth; although strangely enough, they usually make it, by careful design :-)

Stage and a half designs have reliability advantages in that all engines are already lit at takeoff, so there is less chance of a failure in the air- as most engine failures happen at ignition.

The disadvantage is that rocket nozzles work best when they deliver hot gas at the ambient pressure- since as the rocket climbs the pressure changes, so the nozzle of the sustainer engine has to be something of a compromise and loses some efficiency.

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