Click on the "Alliance" menu in the lower right corner. Several alliances will be suggested to you. In case you already know which alliance you want to join, you can search for it by name. Private alliances may invite you to join them, those invitations can be read here as well. Also, you can create your own alliance.

If you have chosen an alliance and have clicked on it, there are two options: Either you can just join that alliance, or, in case it is private, you have to apply to that alliance. In the latter case you won't be able to join the alliance unless they accept your application.