How do spacecraft move? It may seem like a simple question to answer, but it is something actually difficult to imagine. All earthly forms of movement rely on the same principle: Newton’s third law. What this law basically states is that all interactions between systems, and thus all forms of movement, are “symmetric”.
Imagine, for example, an ice skater with a wall to his right. It is evident that if he pushes the wall to the right, he will move to the left. According to Newton’s third law, this happens because pushing the wall to the right is equivalent to being pushed to the left by the wall. To every action there is always opposed an equal reaction: a skater pushing a wall in one direction will result in the skater being pushed by the wall in the opposite direction.
This principle is used for the vast majority of forms of locomotion in Earth: all rolling vehicles are a good example, they push the Earth in one direction so as to move in the other direction (for the infinitely more massive Earth, there will be no noticeable effect). Airplane engines also work the same way: they ingest air at a speed and accelerate it backwards, thus generating forward thrust. Note that the thrust will be generated with respect to the air: the motion of the airplane relative to the earth will also depend on the movement of the air. That is the reason why flights are longer when there is headwind.
In a nutshell: all earthly forms of propulsion rely on interacting with some matter to move relative to that matter. But in the vacuum of space, there is no matter (strictly speaking, almost no matter), so, how do space vehicles move? The only possibility left is for the vehicle to carry that matter within itself and eject it in one direction. The vehicle will exert a force on that mass to eject it to one direction, so it will move in the opposite direction. That is the principle of rocket propulsion: ejecting mass at high speeds to create thrust.
For example, the Russian Soyuz rocket uses a mixture of liquid oxygen and a refined rocket kerosene (respectively called LOX and RP1). Both propellants are combined and ignited. The resulting energy is used to accelerate the products of the ignition at very high speeds (about 3 km/s). But this is only one particular type of rocket using two particular fuels to achieve a particular mission. Space propulsion methods are mightily diverse. Some of the most relevant topics and concepts of space propulsion will be addressed in a series of articles, here, at techforspace.com