Broadly speaking, gravity is the force that causes an object to attract another. The more massive an object is, the larger its pull of gravity. The actual cause of gravity is still debated; it is possible gravity is not actually a force, but it may be caused by curves in space-time. Modern theories on gravity started with Galileo, were codified by Newton and were changed dramatically when Einstein showed that gravity acts contradictory to expectations at small and large scales. However, for the purpose of flying a rocket, Newton's laws accurately describe the force of gravity.
Newton's First Law of Motion says that an object at rest or motion will remain at rest or in motion until another force acts on it. Thus, a rocket needs some sort of exterior force acting on it for it to escape the pull of the Earth's gravity. Once a rocket has escaped Earth's gravity and reached a certain speed, the rocket will remain at this speed until another force acts on it; thus, rockets in deep space (like probes sent beyond Mars) do not need fuel to continue moving unless they are drawn into the gravitational pull of another object.
Newton's Second Law states that force is equal to mass times acceleration (F = ma). Thus, the more massive or the faster an object is moving, the stronger its force. For rocketry, this is important because it means that a rocket must maintain either a high mass or a high acceleration in order to escape the Earth's gravity. A rocket's mass starts off high, but as its propellants are burned off, it gets lighter; thus, as propellants are burned, the rocket must be moving faster (and, indeed, the purpose of burning propellants in a rocket is to get to a point of acceleration that increases the rocket's force).
Newton's Third Law states that for every action, there is an opposite and equal reaction. For gravity, this means that the force of gravity is just as strong on the object that is pulling something as it is on that which is being pulled. While this is negligible for a smaller object (a human being exerting their force on the Earth has little consequence for the Earth), it is important for rockets because it means the force coming from the rocket's engines must be greater than the rocket's mass in order for it to move.