'Around' indicates that something moves in a circular motion, or from one place to the next. For example, Peter walked around the room helping students. 'About' is used to state an approximate amount or number - University costs about $50,000 per year at a private school. That's crazy!

Examples
Drive around the block and pick up Tim.
I'd like to spend about an hour relaxing.
She flew around the US on vacation last year.
It'll cost about $300 to fix your refrigerator.