How can you accept alliance application or invite someone to your alliance?

You need to be R4 or R5 to invite and accept players to your alliance. To invite them go to the alliance menu and tap the Member tab at the top. If you are R4/5 you will see an invite option at the bottom on the screen. Tap this and search for the players name.

Once they apply, if the alliance is set to open recruitment then they can join immediately. If not then they will apply and you will see them listed in the same member screen under the 'New Member' section. You can accept their application and let them join.