Actually its 90 min cycles. (the 90 min is average, it varies per person)
If you go to sleep and use no alarm, you will awake naturaly at the end of one of the 90 min cycles. If you get awoken, alarm or exterior noise, significant other smacking you, whatever, you awake and feel like crap, like you either have over or under slept. Due to the fact you never finished your cycle.
If you awake at the end of one of the cycles, you feel well rested, almost regardless to the number of cycles you actually got, as long as its a multiple of the 90 min.
There are ways to achieve it. Dual alaram, alarm clocks. On set to a low noise so only when your at the light end of a cycle will it wake you, set 15 min ahead of when you want up. So it goes off, then as you reach the end of a cycle it becomes enough to wake you, and your up, feeling good. The second alarm is set loud and at the absolute latest you can get up, just in case
There are also many alarms that use light, no noise to wake up. These work on the same principle and will wake you at the end of a cycle.
Its why sometimes you wake up feeling great with only 6 hours sleep, and sometimes you wake with 8 and feel like crap.
The original poster is taling about polyphasic sleep though, which is related but slightly different. There is a blog site with the daily ramblings of someone doing it at
http://www.kuro5hin.org/story/2002/4/15/103358/720
http://www.ubersleep.com/