When you monkey around, you waste time. When you monkey around with something, you often fiddle with it in an attempt to fix it or figure it out. You might, for example, monkey around with your dad's broken computer.
If a mother who's heading off to work warns her teenager, "Don't just monkey around today," she means that she wants her child to accomplish something worthwhile. If you monkey around, you might do a few random tasks, but mostly you're just fooling around. The expression comes from the way monkeys play: they don't accomplish much, but they have a lot of fun.