Is it in general a security problem not to update your browser. Firefox constantly prompts me to update my browser, but how dangerous is it to not update?

As part of this question, I would like to know what that problem exactly is. What are the risks of not updating your browser? What exactly could happen?

Because there are security vulnerabilities found in software all the time.  These vulnerabilities are sometimes publicly disclosed, sometimes not.  Either way, as developers find or find out about them they patch them.  Running old versions of browsers leaves you vulnerable to malicious websites trying to infect your computer.
Below are links to web pages listing vulnerabilities that have been fixed in relatively recent versions of the 3 most popular browsers.
Microsoft Internet Explorer
Mozilla Firefox
Google Chrome
All browsers are going to have bugs, and all of them will have vulnerabilities.  But staying on top of known vulnerabilities can help prevent attackers from gaining access to your system.
Edit
Thanks to kirb for these extra links to up-to-date blogs of browser security updates
IEBlog
Google Chrome Releases
