The grandfather paradox: you go back in time and kill your grandfather, which means you were never born and couldn't have gone back and committed the murder in the first place.
This paradox is easily explained if you keep in mind that the universe comprises matter and energy, and that space and time are only ways to describe the current state of matter and energy, not corporeal entities in their own right. In that context, time is not a river flowing in only one direction. It, like space, is merely a way of specifying the arrangement of matter and energy in one of the universe's infinite states. (And btw, if you choose a system of units in which the speed of light = 1, then E = M, because matter and energy are actually the same thing.)
So you and your grandfather are collections of matter and energy, little bits of the universe's total supply. In one state of the universe, he's alive and you haven't been born. In another state, he's dead and gone and you're alive. In a third state, the two of you are in the same place and you're murdering him.
The seeming paradox arises when you think of time as that simple stream, moving only in one direction. The universe is in fact a limitless collection of individual moments in which every quanta existent occupies a particular spot defined by time, space, and physical state coordinates. It's as if the universe were a giant, static, space-time cube or matrix. If we had the perspective, we could see that every possible position of every quanta is present; this means, as far as we are concerned, that every possible thing that could happen has happened and hasn't happened, will happen and won't happen, going both forward and backward in time - that is, every state in some sense is there already.
What this means is that pretty much any sci-fi timetravel plot ever contrived is OK - the plot for Primer, for example.