First, select the tournament you want to register for from the lobby list. To the right of the list, you'll see an info box with details about the tournament and a big "Register" button.
When you click "Register" you'll see a confirmation popup where you can click "Sounds Good" to complete your registration, or "No Thanks" if you've changed your mind.
Once registered, you'll have the option to unregister until two minutes before start time. To do this, you can either click the "Unregister" button from the main lobby info box:
Or from the individual tournament lobby window:
You can keep track of all the tournaments you're currently registered in under "My Games & Tournaments" on the right side of the Dashboard.