What's the issue?
Ubuntu updates its software versions every six months with a new release. Mozilla, however, upgrades Firefox every month or two--sometimes every week--with security patches or new feature-based releases. Ubuntu often includes security patches with its own versions a day or a week after Mozilla releases their patches, but many Ubuntu users are impatient and would prefer to install the latest version of Firefox from Mozilla and use that instead of the Ubuntu build of Firefox.
Basically almost all of the methods outlined below do essentially the same thing--download Firefox from the Mozilla website and use that instead of Ubuntu's Firefox. Most also "install" Firefox to the /opt directory.
What are the methods?
Manual - This involves copying and pasting a lot of terminal commands to get Firefox fully integrated with your system. If you want to learn some terminal commands and get to know your system a little better, this is a great way to install the latest Firefox. Instructions are here:
Automated - This script basically takes the commands in the manual version and saves you the trouble of copying and pasting them all. It also automates a few other things--it automatically detects and downloads the latest version of Firefox, it allows you to select your locale, and it verifies the integrity of the download.
Manual with no integration - All the aforementioned methods assume you want the command firefox to launch the new Firefox from Mozilla, that you want your plugins to work with Firefox, and that you want Firefox to be "installed" systemwide for all users. If you just want Firefox to be able to run but not be integrated with plugins or run with the regular firefox command or able to be run by all users easily, then just download the Firefox .tar.gz file from Mozilla to your desktop or home folder, double-click it (or single-click if you're using Kubuntu), extract the .tar.gz using File-Roller or Ark, go to the newly created folder and double-click (or single-click for Kubuntu) the file called firefox. That will launch Firefox.
Use Klik - Klik is a way to install software directly off the web but not using your regular package manager.
First learn how to use Klik:
Then get Firefox from Klik:
Stick with the Ubuntu version of Firefox - Ubuntu will continue to update Firefox with security patches as long as they support the release you're using (Dapper, as of this writing, is supposed to be supported for three years). The security patches may not come immediately when Mozilla releases them, but they usually come within a week.
Edit: Actually, since you are not the root user on your system, the Check for Updates button in Firefox will be greyed out, even if you use the Mozilla version. For more details on how to install upgrades, see this post.
A combination of the manual and manual with no integration methods can be found here.
Why this thread?
There were way too many HowTos on installing the latest Firefox, so I'm creating the ultimate HowTo that outlines the major methods. It's better for new users to have one place to find all the major ways to upgrade than fifteen different HowTos of varying quality. Any "new" upgrade-to-Firefox HowTos will simply be merged into this thread. If a method is actually new and not just a rehash of another method, I'll incorporate that new method into this initial post.