There are a number of reasons your account gets signed in automatically, so try this fixes..
1. Clear the cookies currently residing on ur hard drive manually through the web browsers or Download n install CCleaner from "http://www.microsoft.com/security_essentials/"..It’s a windows cleaning utility..
2. If u r using Firefox browser then navigate to "EDIT > PREFERENCES > click on SECURITY tab > SAVED PASSWORDS (Enter the master password if asked n u know it), then scroll down, Locate n select the facebook links and click REMOVE…This should solve ur username n password auto showing up when facebook page is loaded.
3. if u prefer Chrome…Click on the SETTINGS (spanner) icon > PREFERENCES > PERSONAL STUFF tab > SHOW SAVED PASSWORDS > Locate n select the facebook links and click REMOVE…This should solve ur username n password auto showing up when facebook page is loaded.
4. I’d also recommend u disable browsers offering to save your passwords but then its a shared system and others might want such a function (so discuss wit em)
5. Before Logging in, Make sure the checkbox "Keep me logged in" is unchecked n always click n wait for the successful facebook logout page whenever u r done with d system.

Hope dis helps n Selecting as best answer motivates me 2 helps others.

Easy enough:

Account > Log out

When you log out at the top will be your "email" and password fields. Under that is a check box for "keep me logged in" . Make sure this is not checked.

anytime you start the browser and goto facebook it will prompt you

If you happened to let your browser save your login information, you’ll have to clear it.
How to clear those lists can be found in your browsers help section.