Let me first give you a brief idea of my understanding of Tension force: Assuming a string is made up of a line of molecules, we fix one end of the string and pull it at the other. Each molecule will be pulled in the direction of the applied force and since nothing is moving and assuming the string is somehow made unbreakable, we conclude ( using Newton' laws ) that each molecule experiences a reaction force in the direction opposite to the pull which we call Tension in the string. Note here that the direction of this reaction force is opposite to the pull at every point on the string.
Now consider a situation where we have a string pulled at both its ends with two equal forces. Applying the above reasoning, due to the force at the left end, a tension should develop towards the right and similarly due to the pull at right end a tension along the left.
Now the situation we have is that each molecule is experiencing two tension forces of equal magnitude and in opposite directions which seems to me very absurd. In this particular case of two equal forces at two ends of a string what will be the direction of tension force at any point of the string?