Ubuntu is an unsupported operating system for the BeagleBone Black (BBB), but does have quite a few users, and a stable image with the 3.8 Linux kernel.
To start with, download the latest version of Ubuntu for the BeagleBone Black. You can find the latest version at the Ubuntu software support page. Choose the file that is compatible with flashing to the eMMC. Once you've downloaded the file, you'll want to jump to the section for your specific OS (Mac OS X, Linux, Windows).
It is important that you download a version of Ubuntu specifically built for the BeagleBone Black unless you're an expert at configuring Linux and working with the kernel. It's not necessary to download the most recent version, but it's highly recommended. The BBB ecosystem is rapidly changing, and each release offers many new improvements.
If you've used the the image to flash Ubuntu to the eMMC, the default username for Ubuntu 13.04 (Raring Ringtail) is "ubuntu". The default password is 'temppwd'. Ubuntu does not come with the avahi daemon pre-installed, so you'll need to ssh in using the IP address using your favorite terminal: