I recently started using Aurora and I have been impressed with how stable it was. I was more impressed with how well the Firefox update stream is laid out. Below I have a description of each web browser that Mozilla releases along with the Firefox Beta PPA, Aurora PPA, and the nightly PPA for Ubuntu. To install any of these browsers, add the PPA and do an an update and upgrade.
Firefox, Firefox Beta (referred to here as "Beta"), Aurora, and Nightly are four web browsers developed by Mozilla and the community. In the end they will all become Firefox, but they are laid out like a chain in a brilliant update stream. Each of the browsers serves a different purpose and has a varying level of usability. Here is the breakdown of what each web browser does.
Firefox is the standard web browser sponsored by Mozilla. Additions are added to Firefox only after they have been added to all of the other web browsers in the chain. Firefox is stable and has the most support for add-ons and themes.
Beta is for features that have gone through alpha and pre-alpha testing and are almost ready to be added to the Firefox web browser. This is one last testing phase before additions are finalized and implemented in Firefox. Stability isn't usually an issue at this stage, but some addons with poor support may be incompatible.
sudo add-apt-repository ppa:mozillateam/firefox-next
Aurora is equal to an Alpha stage of development. At this point features are still relatively new, but have been confirmed to be useful and usable by nightly. This is the most stable of the "Cutting-Edge" browsers, but can still be hit by stability issues like crashes with any new features, although it is uncommon. Addon support and Theme support take a much larger hit than in Firefox Beta. Only web developers and computer hobbyists should use Aurora. Normal computer users probably wouldn't find many problems with aurora, but there would be no benefits worth risking crashes and a lack of addons.
sudo add-apt-repository ppa:ubuntu-mozilla-daily/firefox-aurora
Nightly is a web developer where features are added literally every day and then trickle downstream. Nightly users get the raw code churned out from code contributors, which can lead to crashes and problems. Using Nightly is seen as a show of status, almost like a volunteer. Nightly users receive the worst addon and theme support out of the bunch of the builds. A knowledge of programming and patch submission is not required, but is helpful to further the progression of updates inlcluded in this browser.
sudo add-apt repository ppa:ubuntu-mozilla-daily/ppa
Although I mentioned that addon and theming support gets worse as the versions get larger, this is not a dealbreaker. Many good addons are supported on all versions. Nightly is the only browser that may go dark for certain addons.