Newton's third law states that "To every action, there is an equal and opposite reaction" The wording is so crisp that it has become a part of the ordinary language.
It means that forces in nature always occur in between pairs of objects. Force on A due to B is equal and opposite to force on B due to A. Since, they work on opposite bodies, they can't be cancelled out.
Don't add up such forces and claim that the net force is zero : It's an error.
For example, when you walk on the ground, the force you exert on it is equal and opposite to the force it exerts on you, and that's why you accelerate. There is no cancellation of forces. The internal forces within a body however do sum up to zero.
One more thing to be noted is that, the words "action" and "reaction" is somewhat misleading. There is no cause-effect relationship between a pair of forces. Action doesn't precede and cause reaction. They act simultaneously on different bodies.
Another example I love giving in order to explain the concept of internal forces is,
You can lift a chair with your hands, and then it is such that you and the chair are different bodies and mutual forces are not cancelled.
But, what if you are sitting on that chair and without using your feet to get up, can you lift it up ?
You certainly can't, and in that case you and the chair are a part of the same system.
Did you get what I meant ?