How do I add/remove tables to a Windows Azure database?
The Manage button is correct. It should bring up a page in a new window/tab that prompts for the user name and password to connect to the database. If that isn't working, there's something wrong with your web browser configuration (does it allow popups from the Azure site?).

Alternatively, the database server can be set up to allow external connections by managing the firewall rules. The interface makes it very easy to poke a hole just for your client computer. You can then connect to the database directly using Management Studio on your local machine. (Don't forget to delete the firewall rule when you're finished.)