To view your messages, simply tap on the 'Messages' tab in the bottom navigation bar.
Interface Overview
When you tap on Messages, you are immediately brought to your main inbox.
Inbox
The inbox holds all of your current incoming messages. Any messages that have been sent to you, and any invites you've received will live here. They will be displayed in the order in which you received them, and when a new reply to an already existing thread comes in, that message will be bumped to the top of the list.
New Messages and replies are highlighted with bold text, and a colored circle badge, this way you can easily tell which messages have and have not been seen by you.
Once viewed, the highlight will be removed and the message will appear in plain text like all others.
Archived
Tap on the back button to view your Messaging folders and select 'Archived'.
The archived folder includes all of the messages you've decluttered from your inbox. We do not allow users to delete messages, only archive. Rather than worry about accidentally deleting important messages, users can simply retrieve archived messages from their archived folder and move them back in the inbox if desired.
To learn more about archiving a message, please see Archiving a Message in iOS.