To mute a conversation in Facebook Messenger, open the chat, tap the name or information icon at the top, and look for the mute option. You can usually choose whether to mute notifications for a set time or until you turn them back on. This is helpful if you want to keep the conversation but stop alerts for a while.
To archive a conversation, go back to the main chat list and find the conversation you want to hide. On many versions, you can press and hold the chat, then choose archive. Archiving removes it from your main inbox without deleting it, so you can bring it back later by sending a new message or by finding it in the archived section.
If the options are not in the place described, the interface may be slightly different on your device or app version. In that case, look for a menu, three dot icon, or conversation settings on the current page. The wording can also vary a little between mobile and desktop.