If you have not disabled Facebook chat, then you may notice few icons next to friend names in Facebook chat window. Green dots and half moon icons are two types of symbols that may appear in Facebook chat window. Each symbol has respective meaning indicating the status of specific friend listed in the chat window.
Meaning of green dot & half moon icon in Chat box
1. Green dot will appear next to a friend who is online and available to chat.
2. A moon icon will appear next to a friend who is currently idle. Friends are considered idle when they have not taken any action on the site in the last 10 minutes.
Also note, a red notification will appear on the Facebook Chat icon (even if you are offline). This happens when people take actions on Facebook related to you, like commenting on a photo of you or posting on your Wall.