Nearly all propulsion technologies that scientists have developed so far rely on Newton's third law, which states that for every action, there is an equal and opposite reaction. Basically, what this law is saying is that if you push on an object with a certain amount of force, that object pushes back on you with the same amount of force but in the opposite direction. To understand the consequences of this law, we have to examine Newton's first two laws and the concepts of force, impulse, and momentum.
So what does all of this have to do with propulsion? Good question! Suppose you're floating out in space holding a rock. Your mass is about 60 kilograms and the rock's mass is about 5 kilograms. You and the rock make up an isolated system, and so, we can use the above formula for momentum. No matter what happens within an isolated system, the momentum must remain constant. Initially, you and the rock aren't going anywhere, so your momentum is zero and must always be zero (unless some outside force changes that, but then the system wouldn't be isolated). Now, suppose you throw the rock away from you with a velocity of 10 m/s. Then your velocity must also change, otherwise the overall momentum of the system would not be zero. Using the formula for the overall momentum of the system, we can calculate your velocity:
m1v1 + m2v2 = 0
(60)(v1) + (5)(10) = 0
60v1 = -50
v1 = -0.83 m/s
The negative sign indicates that you are moving in the opposite direction of the rock. We could have
chosen the direction of the rock to be negative (so v2 = -10) and then your velocity would have been positive.
The important thing to note is that although the system's momentum stayed the same, you managed to get yourself moving by throwing away the rock. Had the rock been more massive or had you thrown it faster, your resulting velocity would also have been faster. This is basically what Newton was saying in his third law. When you throw the rock, you exert a certain force on it, causing it to accelerate for a period of time. At the same time, the rock pushes back on you with the same force, causing you to accelerate in the opposite direction.
Now, suppose you started out with 10 rocks of mass 5 kilograms. If you threw one rock after another, you would gradually build up your speed. This is the same principle used in most propulsion systems to get something moving. However, instead of throwing rocks, propulsion systems usually shoot out large amounts of gas (known as the fuel or propellant) at very high velocities (similar to if you inflated a balloon then let go, except much more powerful).
