Do you have your browser windows set to open full screen? If so, click on the double box next to the red x to make it windowed, then continue as normal. See what happens when you click on the X. The window should definitely close.
If the windows are not closing it is probably being caused by an Add-on, so try starting Internet Explorer without Add-ons.
Click on Start, All Programs, Accessories, System Tools, Internet Explorer (no Add-ons). See if that window closes okay. If yes then use Tools / Manage Add-ons to eliminate a few at a time until you find the one causing the problem.