It seems to be an issue with some browsers that the indication of an alliance invitation is displayed, but it is not visible anywhere else. In this case you have to know what alliance sent you the invite.


Once you know which alliance it is you can go through the ranking tab and search for the alliance.







Once you find it you will have to open it. There is an occasional glitch where the alliance does not open when clicked after searching. If you open and close the alliance above it then it will work properly.




If you have been invited there should be a "join" button under their logo above the Battle Stats button.

Otherwise you will see an apply button if the alliance is accepting members.




If you are in game with a person that invites you to their alliance you can alternatively get to their stats page through the diplomacy menu. Click the "i" button on their line of diplomacy, and then click their name again to show stats.
From the players stats the name of their alliance is clickable.








The leader of the alliance has three settings for accepting members.



"Invite only" seems to give more difficulty to users trying to apply than the "application must be accepted" setting from the questions I have answered.