You can either join a random alliance from the list or choose a specific one if you know its name.
The Join tab in the Join an Alliance menu shows random alliances. Some of them might be full already, while some of them have certain requirements which you haven't met yet. Check the tab from time to time to find the alliance you need. When you see one you can join, send a request and wait for some time till the owner accepts it or not.
You can also enter a specific alliance name in the Search tab of the Alliance menu. Visit our Facebook community and check comments: players often leave their alliance names there.
Besides, you can always start your own alliance and ask other players to join it.