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This is a very general question.
There are a wide variety of spaceship engines which utilise a wide range of differing mechanisms in which to propel a spacecraft.
In general spaceships engines 'work' by using Newtons Third Law which states:
For every action, there is an equal and opposite re-action.
This means that if a spaceship engine produces an exhaust that leaves the spaceship, then the spaceship's engine and hence the spaceship will be pushed in the opposite direction.
The American Space Shuttle's main engines and its RCS (reaction control system) operate in this manner.
There are some variation in spaceship engine designs, for example Project Orion was a nuclear engine design which would fire nuclear bombs out of the rear of the spaceship (0.15 KT bombs for the 4000 ton spaceship design).
Whilst the firing of the nuclear bomb out of the back of the spaceship would provide a tiny bit of thrust, the majority of this engines thrust was provided when the bomb exploded 200 feet from the spaceship.
The blast wave from the explosion would hit the rear of the spaceship on a specially designed 'pusher plate' and the force of the plasma wave hitting this pusher plate moved the spaceship forward.
Project Orion never proceeded beyond a few small chemical explosive based prototypes due to concerns over the use of nuclear weapons inside the Earths atmosphere but I have included as an example of how varied some spaceship engines can be.
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