How do you invite people to your alliance?

I believe you can only send a specific invitation message if they are not currently in an alliance, although you can send a private message to anyone if you want to talk to them. You will also need to be an R4 or R5 alliance member.

If you want to send a default invitation message then you need to locate their base in the map and tap on it. Then select the Info option. In the Info screen there is a button for 'Alliance' If they are in an alliance this will show you the alliance details. If they are not in an alliance then it will let you send them an invitation message.