I've never seen this kind of thing before: I noticed that the Flash plugin that Debian installs from the flashplugin-nonfree package hasn't updated in quite some time. I've been stuck at Adobe Flash 10.2 for a long while.
I wouldn't have noticed except that Google Chrome is complaining about out-of-date Flash (and Java, too -- but that's too complicated in my case to take care of so quickly).
(Note: This could have something to do with the vagaries of 64-bit Linux support for the Flash player from Adobe. For all I know, this isn't a problem in 32-bit Debian Squeeze. All I know is that I run 64-bit and it's a problem for me.)
I checked Debian's package repository, and my flashplugin-nonfree is indeed up to date. But my actual Flash binary from Adobe is not.
A minute or so in Google led me to this extremely helpful exchange from the Debian User Forums.
It turns out that updating your Flash Player is easy. Open a terminal and either su to root, or use sudo (my preferred method for just about anything rootly):
$ sudo update-flashplugin-nonfree --install
or (with su to root):
# update-flashplugin-nonfree --install
Your system will go to Adobe, download a new Flash plugin for Linux, then install it on your system. That's it!