Facebook Chat is extremely convenient for talking to most of your friends without having to open multiple chat applications. The inconvenient part of Facebook Chat is that if you leave it open, your friends know the minute you log onto Facebook. If you want to be able to log on in peace and not necessarily let your friends know, you need to change your chat status.
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Navigate to Facebook.com and log into your account.
Click the Chat window in the bottom-right corner. When logged in, you should see a checkmark next to "Available to Chat."
Click "Available to Chat" to mark yourself as unavailable. You now appear offline. If you want to close the Facebook Chat window, click "Hide Sidebar."