The red line above is a cut through the wood. It can be either a saw cut or a split. In any case, if it precisely follows the actual grain, there is no runout. Wood being wood, however, the ideal situation sometimes does not occur.

If the cut is a bit off from that natural grain, as in the sketch above, it reveals what is known as "runout." This sort of thing happens when you saw the wood. It's nearly impossible to do by splitting or riving the wood. Even if it is sawn, it is still cut on the quarter, but the result is that the light striking the wood will reflect differently.
 

The wedge at right is what's known as quartered, or quartersawn.

If you split this wedge precisely down the center and open the resulting two pieces like pages of a book, such that the outside face of each wedge is pressed together, the pieces are said to be bookmatched. Like this:

The narrower edges at the outside of the top were once in the center of the tree. 
 

Now, here are some images and accompanying text from Frank Ford's FRETS.COM Illustrated Glossary, with a bit of my commentary (in brown) farther down.
 
Frank: The tricky part is that sometimes it's not all that easy to see in a piece of lumber, and the grain may run perpendicular to one face, and have a lot of runout in a perpendicular face of the same piece of wood.

Here's a piece of spruce:

It's easy to see how the grain runs in this dimension. It's quite straight and true, just like an ideal instrument top.
 

This is what is called "vertical grain." You can see from this cut end that the grain lines are almost straight up and down in relation to the flat face of the board. There's no question how this wood will split if I break it between these grain lines. It will split directly from one end to another right along the grain. 
 

The long knife is a Japanese bamboo splitting tool*. It's very thick and quite sharp, and it's the perfect thing for splitting small billets of wood. The other tool is, of course, my heavy wood carver's mallet. 

(*This tool's English name is "froe." And to cut with it is "to rive," hence the term "riftsawn," which means sawn along the natural line that could have been riven or split.)
 

You can't really tell the grain direction along this face, even if you look really closely.
 

Now, it's really easy to see the grain direction:

No runout at all! The grain in this piece runs parallel to the face and the piece splits neatly into two flat pieces of uniform thickness. 

This is, by the way, how the very finest instrument wood is cut. To make absolutely sure that there's no grain runout, the tops of instruments are sometimes cut from "split billets" to ensure straight, uniform grain with no runout. 

Whoops! Clear runout here:

I couldn't tell that there was grain runout by looking at the edges, but spitting is a dead giveaway.
 

Each side will reflect light differently, giving the same effect as wood that's not riftsawn, or which has runout.
 

Well, for one thing, wood with no runout is stronger. If it is really stressed hard it won't break as easily because the grain can't split off the edge. Severe grain runout is bound to have an effect on the vibrational characteristics of a top as well as an effect on the strength. I don't pretend to be able to discuss the theoretical differences in tone, but in a few cases I've actually seen tops fail structurally because of grain runout. In those few instruments, the grain runout was astoundingly bad, say around 30 degrees off parallel.

The guitar pictured above happens to sound GREAT, and shows absolutely no structural difficulties despite its 29 years (as of 2005) under string tension.

The matter of grain runout is largely a cosmetic one, and most builders try to avoid runout if possible.