Muting a conversation simply will ignore the messages being sent to you.
What muting a Facebook conversation can do?
Muting a conversation will keep chat tabs from popping up on your Facebook page and will also turn off push notifications on your mobile as well. In short, you won't get any alerts or notifications whenever a new reply is written.
If you're too busy or not in the mood to chat with any person, you can simply mute the conversation to stop getting new message alerts.
To do this, click the option button, which can be found between the “Video Chat” (or "Add People" icon) and “Close” button. When you click the "Option" button, a drop-down menu will appear. Find the “Mute Conversation” option and you will not receive any more messages from that person. Here's a sample screenshot:
I really find this "mute" feature very useful especially when my contacts include me in a group message that has nothing to do with me, which can be pretty annoying. So muting conversation is the best way to go.
Some other questions..
Am I still be able to view the muted thread? Yes, you will still be able to see your friend's response(s). Just view your inbox and you should be able to find the muted conversation.
Will muting a conversation delete the new messages (replies)? No, you will still receive new messages, you won't just get notified about the new replies from the muted thread.
What happens to your muted messages?
Well, just in case you become curious what you missed, you can check out your archive chat to find the muted messages. They are not really deleted but they only stop appearing from your chat box.
What if.. I want to un-mute amessage?
Go to the messages section/page, search for the muted thread and you will see a "Unmute" link at the upper right corner of the conversation.