The firefox-stable repository has been recently abandoned, and right now the whole system is undergoing changes. Apparently they are going to include the latest versions of Firefox in the Ubuntu official repositories (they should have since January) but unfortunately according to the OP that hasn't happened yet. Right now, I can't easily find a way to install Firefox version 10 on Ubuntu because of the recent change.
However, I found a PPA that is actively maintained for Firefox version 11, which might be cool because you'll be running beta software (not necesarily bleeding edge
) and might get speed benefits and such. Just make sure you remove your old PPA (open up Software Sources). After open up a terminal and do this:
sudo add-apt-repository ppa:mozillateam/firefox-next
sudo apt-get update
sudo apt-get install firefox
This might all seem confusing to you, but what all major Linux distributions do is that they maintain their own versions of software and put these in repositories. These repositories are available for everyone who use the operating system, and because of that installing software is as simple as running a simple command. Problem is that the software they maintain might be out of date from what actually exists upstream, like in this case with Firefox. That is why PPA's exist in Ubuntu/Debian, which are small repository extensions that you add to install additional software, or software that might be more (or less) up to date.
I would like to also say that it is extremely unusual to install software with installers in Linux. Because the package manager and repository exist, there is no reason to try to install software outside of the repository. This is a key difference between Windows and major Linux distributions.