It means you can chat to people on Facebook who are your friends and are online when you are.
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Add people as friends, and when they are on facebook, they will appear in the Chat menu in the bar at the bottom. Click on someone's name to chat with them.
It cannot be deleted, but it can be hidden. Click on the settings icon (looks like a mechanical cog) and select "hide". If you wish not to be contacted in facebook chat, then deselect "available to chat".
3 does not meant for anything on Facebook chat. It's just a digit.
Sometimes the Facebook chat can disappear due to a glitch with Facebook. Try closing out Facebook and clearing your browser cache and then restart the application.
You spelt messenger wrong. And you don't 'do' images on Facebook Chat (there's no such thing as Facebook Messenger, I assume you are referring to Facebook Chat) as it is impossible. Email images, post them on Facebook, send them in the post. On Facebook Chat, you can only have conversations. Just get MSN. Idiot.