Browsers, just like any other programs have bugs. Most bugs are just annoying. But some bugs can be exploited to violate the security of your computer. You browser receives and processes data sent to it by web sites. A web site can send your browser some data that exploits a bug in a browser and violates your computer's security.
Different security bugs can allow different attacks. Some bugs make the browser download and run a program supplied by a web site without asking your permission and without giving you a warning. In this case the attacker can do anything to your computer - erase your files, infect you with a virus, install remote control software, snoop on your emails, etc. Other bugs are less disasterous. They can allow, for example, to read portions of some files on your computer.
Just login to this website and test your browser before you logon to any other websites to make sure you have a secured browser. http://bcheck.scanit.be/bcheck/index.php