What is "causality" and what does it have to do with time travel?

What do we mean by time travel?

Technically we're all travelling in time just by existing. But we can't seem to control our motion through time in the same way we can control our motion through space. What most people usually mean by time travel is the ability to drive around in time they same way they'd drive around in a city. Let's look at a spacetime diagram that shows a few examples of different ways observers or objects can travel in time and space.
It's important to remember that any single point on this diagram represents an event - both a moment in time and a location in space. The origin of the axes in the figure represents the place X=0 at the time T=0. This is what we mean by a spacetime event.
The figure to the left is a spacetime diagram showing spacetime paths of various observers moving in 1+1 dimensional spacetime. Paths A, B, C and D represent the normal kind of time travel that we find in our world. Path E shows an example of a kind of time travel that is not allowed in Special Relativity.

The allowed time travellers

The light blue spacetime path A represents a flash of light from a laser coming from off of the diagram. The flash of light travels into the future to the spacetime event T=0, X=0 at which it intersects with the spacetime path B.
Path B is a vertical line. A vertical line on this diagram means an observer or object at rest in this coordinate system, staying at the same value of X for all time T. In this example, the green spacetime path B is the worldline of a mirror standing on a table. Let's say that this mirror is only half-silvered, so that when the laser pulse path A intersects with it, half of the pulse is reflected back to the source.
The reflected half of the laser pulse is shown by the light blue spacetime path C. The half of the pulse that is not reflected but transmitted is found by continuing path A.
The purple spacetime path D starts at rest, then accelerates and keeps accelerating until it approaches the speed of light.

The forbidden time travellers

Our abnormal time traveller is shown on the red spacetime path E. Notice this path is a circle. But it isn't like a circle in space -- anyone can walk around a circle in space. This path is a circle in spaceTIME -- this path keeps going back to the same TIME as well as back to the same space.
Notice also that spacetime path E is almost always in two places at the same time.
What's wrong with this spacetime path?
Over 50% of spacetime path E is travelling faster than the speed of light. At the top and bottom of the circle, the observer would have to be travelling infinitely fast.
We'll explore the difference between normal and abnormal time travellers in the next section.