Global chat is separated into invisible clusters, and you'll be assigned a cluster when you load the game. There is no way to choose which cluster you'll be a part of, or which one you're on. This is to improve performance and keep chat activity manageable.
Messages typed in General chat are visible by all online players in your cluster.
Messages typed in Alliance chat are only visible by members of your Alliance.
Once you earn enough VIP points through purchases to reach VIP Level 1, you'll be able to enter VIP chat and speak to other elite players.
By tapping on a user's name and selecting Chat, you can chat privately with them.
Note that private message threads cannot be deleted from your list, but the list will scroll as it grows. If there is no activity in a conversation after a set amount of time has passed, it will be removed from your list. You can re-contact that player manually to resume chatting.