To sign up for an App Engine account, visit http://appengine.google.com. This will (currently) put you on a waiting list so you won't be able to move to the next step for a while.

Once your account is created, you'll receive an email from Google. Now when you visit http://appengine.google.com you'll reach the Welcome screen:



Clicking "Create Application" takes you to this screen:



Be careful choosing an Application Identifier because you cannot change it. You can change the Application Title later (this is apparently only used when users login). The Check Availability button gives a quick way to see if your desired name is available. I left the Authentication Options set to the default.

Once you find a name you like and click Save you will see:




Now whenever you visit http://appengine.google.com you will see your account(s) in the console view:



That's it! The hardest part is waiting for your account to be activated.