Use off and on to describe things that are intermittent or irregular. If you're home sick, taking many short naps all day, you might say you dozed off and on.
When the weather report calls for it to rain off and on all day, you should probably bring an umbrella, even though there will be dry spells from time to time. Something that happens off and on isn't constant or steady, but fluctuates. Off and on is also an adjective: your friends' off and on relationship is unpredictable — you never know if they're dating or not!