If you click on "Activate Adobe Flash" when Firefox warns that "this plugin is vulnerable and should be updated", you can still use Adobe Flash.
What could a website do to you if the "plugin is vulnerable"?
There are several vulnerabilities known in previous versions of Flash which allow to execute arbitrary program code. That means an exploitive website can do anything a program can do which doesn't require administrator privileges, and there are ways to get around that prompt ("privilege escalation").
Among the common exploits which websites like to install through flash vulnerabilities are:
So seriously, keep flash updated, or you are easy prey for criminals.
You might also consider setting flash applets to "click to play" in your browser settings (firefox|chrome). That way a website at least needs to trick you into believing it has some content you are interested in to infect you and can't just do it simply by you visiting it.
The real base question here is
What can a malicious website do if it gains access to a run an application on your computer?
Which further boils down to
What can a malicious application do on your computer
Which is pretty much whatever it wants within it's level of access. If it can elevate it's level of access even further it's no longer your computer anymore. It's their's.