To change your driving directions on the fly, hover your mouse over the point in your route you'd like to change. You'll see a small square which represents a control point in the route. Click on or near the route and drag it to a new path.
Your path is instantly updated, and your driving directions change to reflect the new route.
This is a powerful tool, but it's easy to accidentally overdo it. If you find that you've changed your route too much, you can use the back arrow on your browser to undo the damage.