If an alliance officer sends you an invitation, you can accept the offer (click on the alliance button at top of screen).

Once you have a 1st level Embassy, you can ask to join an alliance through your Embassy.

Joining through your embassy is MUCH easier on the head of the alliance because they can see your might and accept you directly from their embassy.
(instead of having to look you up and add you and then have you accept it later).

Alliances are limited to 100 members and some require a minimum might, recommendation of another member, or frequent play.

You can only be in one alliance at a time.