It seems I am really having a difficult time understanding tension.

I understand that if two people are pulling on opposite ends of a rope each exerting a force of 25 N, the tension of the rope is 25 N, but when the proportions are changed, I admit I start to get confused. What kind of tension would exist in a rope where one person is pulling in his direction with a force of 25 N and the other guy is pulling with a force of 20 N in the opposite direction?

Using my own common sense, I would think that the tension of the rope would be equal to the lesser of the two forces, but aparently that is not correct. Sad

Assuming that the rope is uniform (i.e. every centimeter of it has the same mass), the tension would be a linear function of distance. Greatest at the point where the greater force is applied, gradually decreasing towards the opposite end.
And the rope would be accelerating at a rate equal to \frac{F_1-F_2}{m}, where F_1 and F_2 are the forces and [/tex]m[unparseable or potentially dangerous latex formula]

What kind of tension would exist in a rope where one person is pulling in his direction with a force of 25 N and the other guy is pulling with a force of 20 N in the opposite direction?

It is an impossible situation. You cannot pull with a force of 25 N unless it is resisted by a force of 25 N in the opposite direction.

Just try to break a string that is not anchored at the other end, or does not at least have a weight that can be accelerated by the force you apply. It cannot be done.