Monophasic Sleep is where you sleep in one block during any 24 hour cycle. Polyphasic Sleep is where you sleep more than once, but usually still in a regular pattern. One common example of this is in countries in which it is common to take a siesta: foregoing the unusable part of the day where it is too hot, but staying up later into the night in return.
However there is something strange about Polyphasic Sleep, the more times you sleep, the fewer total hours of sleep you need. So typically people sleep somewhere between 7 and a half and 8 hours a night, but by sleeping 2-3 hours twice a day you will feel fine after the first few days. Some people take it even further and I mentioned in the biography that I wrote about Buckminster Fuller that he slept for half an hour four times a day. This practice he kept up for two years.
When I was at university one of my friends decided to commit to a Biphasic system. After all, a university timetable is very easy to fit odd sleep cycles around and I'd suggest doing that XKCD 28 hour day, 6 day week experiment while you're not bound by the 9 to 5 world of the adults.
My friend's pattern was to sleep for 4.5 hours and wake at the same time as everyone else, then to sleep for a further 1.5 hours from 5pm until 7:30. This gave about 6 hours a day, which is a lot less extreme than some of the schemes that people carry out, but is still a solid two hours of extra awake time per day. He kept up the sleeping pattern for about a month and has said that eventually the natural sleep cycle catches up unless you fight it enough.
The idea is that when you fall asleep you go through a short period of REM before sinking through light sleep and then to deep sleep. You remain here for about an hour and then the cycle through light sleep and REM for the rest of the night, finally ending in mostly just light sleep fully rested. If you have a short amount of sleep, limited to around 20 minutes then you are getting almost exclusively REM, which is the bit which is essential. The deep sleep is good, but the light sleep is mostly useless in practical terms (although great if you love dreaming, which I do).
What I found quite surprising is that sleeping in one long session is a relatively modern convention in Western Europe. Before the industrial revolution most people would wake for about an hour, usually in the 2am to 4am region of the night. If you have ever woken up in the middle of the night to fetch a glass of water and you have experienced a slightly blissful serenity you will understand the main argument for this system. Writers and poets would often write part of their works in this time and most would use it to reflect or pray (or have sex it would seem).
Most Romance Languages have a word or phrase for this awake period, but we don't seem to have an English one. It is unclear why it died out, since it would seem that the invention of electronic lights would make waking in the night easier. However the leading theory is that with the industrial revolution came a more regimented attitude to shift starting times at factories and other places of work and with them the more erratic sleep cycles were forced out.