Trello doesn't have "Favorites" exactly, but you can control which boards show up in your boards menu by "pinning" (or "unpinning") a board
If there is a board that you aren't a member of, but you'd like to keep an eye on (e.g. the Trello Development Board), you can add it to your Boards Menu
- Click the icon next to the board title (or click the Board Profile button on the right)
- Click "Pin to Header Menu"
(The Boards Menu is the list of boards you see when you click the "Boards" link in the upper right)
You can also use that menu to unpin boards that you don't want in your header. (e.g. if you're a member of a board, but you don't check it very frequently)
A list of all of your boards, including the ones that are not pinned, is available on your boards page