I am awesome at taking naps. Just ask any of my former classmates or coworkers or the Jersey barriers on the I-5 and 210 freeways.

What happens when we fall asleep? This process happens in four stages:

0. Transition. Falling asleep in lecture. It is common to experience hypnic jerks.

1. Slow eye movement. People aroused from this stage often believe that they have been fully awake. This is where we experience a sensation of falling. Duration: 1-7 minutes.

2. No eye movement. Body temperature begins to decrease, the sleeper is easily awakened. Duration: 10-25 minutes.

3. Deep sleep. Passed out on my desk. The brain consolidates declarative memory (the ability to recall facts and knowledge). Dreaming can happen, but dreams are disconnected and less memorable than during REM sleep. This is where bed-wetting occurs.
Duration: 20-40 minutes for the first cycle, decreasing to 0 in later cycles.

4. Rapid eye movement (REM). This is the lightest stage of sleep. The brain consolidates spatial memory (the ability to recall information about location and surroundings) and procedural memory (cognitive ability and motor skills). Dreams are vivid, and heart rate is increased.
Duration: 1-5 minutes for the first cycle, increasing in subsequent cycles up to an hour at the end of the night.

After the REM stage, we either wake up or go back to stage 2. We then cycle from stage 2 to 4 until the end of the night, with stage 3 growing shorter and stage 4 longer each cycle.

Waking up from stage 2 or 4 makes us feel refreshed, but waking up from stage 3 leaves us groggy.

In summary:
To improve learned skills and figure out directions better, sleep the whole night. To improve rote memory, take lots of hour-long naps.

See Also:
Natural Patterns of Sleep (Harvard Div of Sleep Medicine)

I was inspired to write this post after reading a WSJ article of similar intent, but inaccurate information.