Here's a slightly more mathematical answer to go with Vortico's excellent physical answer.
Tension is not a force, and doesn't have a magnitude and direction. This is confusing, because we frequently talk about the "force of tension", which is a different thing from the tension itself.
The tension is actually an example of a mathematical object called a "rank-two tensor". A rank two tensor is a function whose input and output are both vectors. The input is a direction, and the output is the force exerted on the piece of rope lying in that direction. So for a vertical rope, you can take a unit vector pointing down as the input to the tension tensor, and it gives back a force of, say, 20N up. That would mean that at any point in the rope, the rope is pulling up on the parts beneath it with 20N of force. Then if you input a unit vector point up, you'll get back a force of 20N down. Any part of the rope pulls down on whatever is above it (more rope, or maybe a ceiling) with 20N of force. We then say that "the tension in the rope is 20N", but what we mean is actually the entire story above about inputting directions and getting out forces. Every little piece of rope has two 20N forces on it from the surrounding rope, and also exerts two 20N forces on the surrounding rope.
This is a little pedantic, since the tension is just a number and the whole bit about the tensor does not seem very important. It becomes more important when we talk about surface tension, perhaps in a metal sheet. In that case we could imagine that the tension really is different in different directions. Physically, if you cut a slit in the sheet, and then measured how much force it takes to hold the slit closed, you could find that the force required (per unit length of the slit) depends on the slit's direction. You would need the full power of the rank-two tensor to describe that tension.