I was reading Discover magazine recently.  One article said that the
universe is about 13.7 billion years old.  Another article said that
one gamma ray burst was determined to be about 12.8 billion light
years away.

Those two articles made me wonder how, in a ?mere? 13.7 billion years,
the universe could have expanded 12.8 billion light years in all
directions, and THEN have the light of the most distant gamma ray
bursts travel all the way back to Earth (i.e. another 12.8 billion
years).

12.8 plus 12.8 equals 25.6 

If the universe is only 13.7 billion years old, how do astronomers
explain the 25.6 billion years needed for some galaxies to have moved
enormous distances and send their gamma rays back to Earth?

That's a really good question and I hope a General Relativist gives us
an authoritative answer. For an object that far away, the curvature of
space will certainly play a role. Part of the answer however, is that
the object in question was not 12.8 billion light years away from us
when the light was emitted. That is the distance now, but the light
was emitted when it was closer to us. To get this right, one has to be
very pedantic about what "time" and "distance" we are talking about.
It helps to draw space-time diagrams to understand this.

The answer, in a word, is inflation.  The theory is that when the
universe was only a tiny fraction of a second old it expanded at much
more than the speed of light for another tiny fraction of a second. 
By doing this, it managed to avoid a lot of the problems with the big
bang such as the one you've highlighted.  Because it was the entire
universe that was expanding, that doesn't violate relativity! The
Wikipedia article is here  -
http://en.wikipedia.org/wiki/Cosmic_inflation .  The theory was first
proposed in the early 1980s by Alan Guth, and somewhere on the web you
can find an audio file of him explaining the theory - it's well worth
a listen!

Ian G.

Inflation makes the question even more interesting, but it is still a
valid question in a universe without inflation.