Assume a rope on a smooth surface. If the rope is pulled from one side (force exerted on one side), where is the equal and opposite force? Why does the rope move? What I mean here is that there is no mass either side of the rope, it's just laying on the ground. I pick it up and then drag it. Shouldn't the force of tension balance out?
Newton's Third Law says that for every force there is an equal and opposite force. However, it must be clear that these forces act on different bodies. For example, if you push a wall, you will feel a force against your hand, which is due to the wall. It is the wall's reaction to the force you are applying to it.
In the specific case of the rope, the equal and opposite force is on the object which causes the force acting on the rope (every force involves more than one body. In Classical Mechanics, a body cannot interact with itself). The rope will move, because there is a resultant force other than zero acting on it and, therefore, there is an acceleration on it, which will cause a velocity change and, finally, motion.
A similar situation is that of a horse pulling a carriage. There is a force backwards on the horse and there is a force forward on the carriage. So how can the system move? It is simple: the forces act on different bodies, so then don't add up to zero.
The equal and opposite force is due to the rope acceleration according to $F=ma$, where $F$ is the force, $m$ is the mass of the rope, and $a$ is the acceleration of the rope.
By pulling the rope, you are exerting a force on it, and so Newton's third law tells us that it too exerts an equal force on you but in the opposite direction. Obviously, this force is not very large, so it isn't enough to accelerate you by any significant amount.