It's the meebo bar. You can chat with your friends in that chat bar.
If you want it gone or dead, go to 'Account', then 'Privacy', then look for chat bar settings. You can choose the settings yourself. So for example, if you want it gone and dead, just click 'permanently delete'. (It should be something like that.)
Did I get a virus or did Chess.com add a an intrusive add bar to my page? I only get it when I connect to this site.