The force of tension acts right to left and is the difference between the pulling force and the resistance forces (air resistance and friction). I dont know what country you're in but in the UK you get taught this if you continue doing physics or maths after you're 16.
I'm not certain what your refering to when you say normal force, but in your scenario, the only normal forces are the force of tension which will act in the direction of the rope (towards the box), and those up from the ground against you and the box.