Azure has been making some news recently since it's partially responsible for the Firefox 7/8 development builds being so much faster than earlier releases of this open-source web-browser. (Along with Azure has been improvements to reduce Gecko's memory foot-print and garbage collector.)
The Azure library was designed to be a state-less graphics API that's closer to the platform / hardware APIs than is the stateful Cairo API. The new library also requires less format conversions for Gecko's usage than does Cairo. Mozilla specifically designed Azure to be very close to the Direct2D API, but still suitable for targeting OpenGL and other low-level graphics APIs. Azure is also better suited to handle the canvas element of HTML5.
A nice introduction to Mozilla Azure can be found on the Mozilla blog from last April, but since then the project has matured a great deal and is ready for Firefox 7 with further refinements in Firefox 8.
The back-ends for the Azure library are Direct2D, Quartz, OpenGL, Direct3D, and Cairo. On Windows, Direct2D is the primary target for Azure while Apple Mac OS X users will have Quartz, and Linux users have OpenGL. If hardware acceleration fails, Azure can fall-back to having Cairo sit underneath. Cairo is also used for Gecko's printing support. Mozilla's copy of Cairo has also been modified to take advantage of Azure's internal stateless surface API. Direct3D 9 and Direct3D 10 back-ends for Azure are also being developed.
On this blog are some benchmarks comparing Cairo and Azure for those interested.
Mozilla's fierce competitor to Firefox, Google Chrome, meanwhile doesn't use either Azure or Cairo. Chrome is using the Skia 2D graphics library, which is also developed by Google and offers support for multiple rendering back-ends.