Sleep cycles, they're incredible. Disclaimer: not a biologist. Roughly every 90 minutes you almost wake up in your sleep, if an alarm wakes you up around this point you are much much less tired than if you wake in the middle of one such cycle. On average it takes 14 minutes to fall asleep.

I use DoubleTwist on my Android phone and it works incredibly, but that's a paid app you don't really need if you do the simple math.. and of course your mileage may vary. :)