Archiving a conversation hides it from your inbox until the next time you chat with that person, while deleting a conversation permanently removes the message history from your inbox.
To archive a conversation:
Tap at the top of your screen
Swipe left on the conversation you want to archive
Tap Archive
To delete individual messages or your full conversation history, open the conversation and tap (at the top-right), then select Delete or Delete Selected. This option is currently available for individual conversations only. To delete a group conversation, you'll need to use Messenger or a computer.