At an operational level, you are simply combining equal powers of 2 for as long as you can. Wikipedia says 2^17 is the absolute maximum, but the odds of doing this are astronomically small.
I've only gotten to 1024 myself, which is pretty low in comparison to what's possible once you master the technique. A friend recently claimed he pulled off 16,384 (= 2^14).
In a nutshell, the strategy (as far as I understand it) is to coalesce your largest numbers in one corner (or just one side), and to line up a row of successively smaller powers to the left of it. So, for example, you might have 32 in the bottom right corner, with 16 to its left, then 8, and then 4, so that the bottom row is filled up. Then the bottom row is fixed so long as you only move right and left.
At this point you should try to get another 4 next to the 4 at the bottom left. Once you do that you can combine all of the bottom row into a 64, since 64 = 2^6 = (2^5 + 2^4 + 2^3 + 2^2 + 2^1) + 2^1. After a bit you can probably get something like this:
...and continue in this manner until the corner number gets bigger and bigger. You can see I did not totally succeed, since the 512 tile isn't in the rightmost square. As you progress it becomes harder and harder to avoid this issue, until you are overwhelmed completely by low number tiles, at which point you lose.
As you go higher, each new power of 2 takes longer, requires a longer chain (in the strategy), and the board gets more and more cramped. Frankly, I don't quite know how to do it, or else my score wouldn't suck so much.
Britlandish