Facebook Chat has a feature which lets its users appear invisible in the IM. So users can chat only with the person they want to chat with. While appearing invisible in Facebook Chat, users still can be online on Facebook IM network.
To appear Invisible or offline to All or some Friends on , follow these steps:
1. First Click on the Chat pane on the bottom right corner of your Facebook page to bring up the Chat window.
2. At the end of the line of your Friend List name you can see a white/green toggle, switch it to appear invisible to every friends inside that list.
When you set a particular Friend List in offline mode, the toggle switch will be displayed in grey/white color.
3. To appear invisible to everyone, click on each and every “Go Offline” toggle, alternately you can click on options and select "Go Offline" as well.
4. If you want to appear online to certain people in a group then you need to toggle the switch back to online for that group. You must keep in mind if you toggle it back, you will appear online to all the people in that group.
It is possible to create a new Friend List consisting of just those people to whom you want to appear online.
To make a new Friend List, click on Friend Lists tab in the Facebook Chat then type in a name for the new group and then drag and drop users to the list.
Alternatively, you can go to the Friends tab, and click on "+ Create a List" button. Enter the new group name, then include friends in the list, and then click on Create List button.
Any changes to the Friend List will be automatically saved.