Here's my shot at it:
Everyone has a "biological clock" triggered by the sun and the approximately 24-hour cycle. When the sun rises, the light enters our retinas and provokes wakefulness. When the sun goes down, and the lights are out, people start to chill and relax and get ready to go to sleep. Additionally, throughout the day we have periods of "clock-dependent alerting," meaning that we get a "shot" of alertness at around 2pm depending on the person, and in the evening if we stay up too late.
If we don't get enough sleep -- and this is a contentious point -- we may accumulate a backlog of lost sleep and feel drowsy for some time. Not all research suggests this is true, however.
Some people like naps and some people don't. A short nap in the afternoon (or a siesta) can either refresh you or prevent sleeping at night, depending on the circumstances.
While asleep, we need to dream to rest our brains. There are four stages of sleep, getting deeper, and REM, the time you dream most, is the second stage. First stage is when you're just dozing off, third is deep sleep, and fourth is REALLY deep sleep.
REM stands for "rapid eye movement," because your eyes are darting around like crazy. If you wake DURING a REM period, you are more likely to remember your dream than if you wake at a different time. It is thought that if you disrupt REM sleep, you are more likely to have a bad day the next one.
Some of us even have lucid dreams. A lucid dream involves being aware that you are dreaming, or even being able to control your dreams. They can range from frightening to hilarious. Research shows that people can actually be TAUGHT to have lucid dreams, and there are some commercially-available eye-pieces that are supposed to alert you of when you are (by sensing your rapid eye movements).
Finally, depression can lead either to broken sleep, little sleep, or excessive but not wholesome sleep. Mania involves very little sleep.