When someone "dozes off," they fall to sleep for a brief period of time. They may not have had the intention of falling asleep, but they did anyway...this often happens when someone is watching TV.
"Taking a nap" is when someone goes and sleeps for an hour or two during the middle of the day. For example, after I am done with classes I am usually pretty tired, so I take a nap around 2pm so I feel refreshed when I wake up