I am very much confused about tension. I have even Googled my doubts (Also used khan academy) to clear my doubts but even there explanation couldn't satisfy me. PLEASE SOLVE THIS QUERY:- what happens to the force that we apply to the rope externally, doesn't it get cancelled out by newtons 3rd law. And how at atomic level the force that we apply is transferred to the other end of the string. It would be of great help if you answer your explanation with diagram since I have my test tomorrow.
It may be a bit difficult to convince you that tension works the way they say without several back and forths. However, your specific questions can be answered, and they may help you figure out what piece of the story you're missing.
Newton's 3rd law does not generate forces that cancel out. Newtons law says that for every force, there is an equal and opposite reactionary force. The key to this (which is often not covered very well in physics classes), is that the reaction force happens to the other object. What's really said by Newtons 3rd law is "if we want to make sense of motion using forces (i.e. F=ma), we must understand that forces come in complementary pairs." I hate the phrasing "opposite" because it causes confusion. They call it "opposite" because the reaction force occurs in the opposite direction, but I find "complementary" captures the idea that one force cannot exist without the other better.
If I push on an object with a force, there must be a complementary force -- the object pushing on me. The more force I apply to the object, the more force is applied back at me.
In the case of your rope, if you have something that's externally applying force to a rope, the reaction force applies back on whatever thing is applying that external force. The reaction force does not apply to the rope. It applies to whatever the other thing is. If we then look at any element of the system, whether it be the rope, or the thing applying the force, or perhaps a pulley that the rope might be stretched over, we can then use F=ma to determine how it is moving.
As for the atomic level forces, the key force you're looking for is electrostatic forces which hold molecules together and push molecules away from each other. When you "pull on a rope," you actually apply electrostatic forces which try to push the rope in the direction you are pulling. And, indeed, they succeed at moving the atoms of the rope, perhaps a few nanometers. As the atoms move at this nanometer scale, they generate surprisingly powerful electrostatic forces to pull atoms further up the rope towards them (and, by newton's third law, those atoms further up the rope apply a force on them, restraining their motion). This propagates mighty fast (at the speed of sound), over very tiny distances (nanometers, micrometers at most). This happens so fast and over such small distances that, for all intents and purposes, we say the rope "moves as one object," permitting us to use F=ma where m is the mass of the rope.
One other fun challenge of learning how tension works is that physics classes love to ignore the mass of the rope. For realistic secnarios, such as using a rope in a tug of war or to lift up a package, this is a very effective simplification. The mass of the rope has very little effect on the final result. However, in theory the mass is there. And, if you feel like it, there are physical exercises you can do which are dependent on the mass of the rope to provide you a good workout. If tension is not making sense, try remembering that every little snippit of the rope does have a mass, and is obeying F=ma as it moves.