Can a Raspberry Pi run Ubuntu?
From the Official FAQ Page at Raspberry Pi:
What Linux distros will be supported at launch?
Fedora, Debian and ArchLinux will be supported from the start. We hope to see support from other distros later. (Because of issues with newer releases of Ubuntu and the ARM processor we are using, Ubuntu can’t commit to support Raspberry Pi at the moment.) You will be able to download distro images from us as soon as the Raspberry Pi is released, and we will also be selling pre-loaded SD cards shortly after release.
Source: My answer on Raspberry Pi SE
EDIT
It seems that since the launch of Raspberry Pi 2, Snappy Ubuntu Core can be run on RP2.
Thanks to a very successful collaboration between the Raspberry Pi Foundation and Canonical, the company behind Ubuntu, we are pleased to announce that Snappy Ubuntu Core is available for the Raspberry Pi 2, the latest Raspberry Pi family member.
- Snappy Ubuntu Core on Raspberry Pi 2 | Ubuntu Insights
However, it should be noted for the anxious that this is not a desktop version of Ubuntu.
Currently not - Canonical only support ARM's ARMv7 processor architecture and the Pi has an ARMv6 chip.
Some complications with ARM are a common issue for all distributions, they are not always Ubuntu-specific. They stem from the fact that ARM has a number of different architectures, offering limited to no compatibility between each other.
The leading Linux ARM distributions are the following (excluding Gentoo which by definition is compiled to the exact target), and I am listing the corresponding target architecture:
Ubuntu: ARMv7 and up, thumb 2 (32 bit)
Debian: ARMv4t and up (armel) ARMv7 and up (armhf)
Fedora: ARMv5tel (arm) ARMv7hl (armhf)
Arch Linux: ARMv5te or higher
To your specific question, the original Raspberry PI runs a BCM2835 (ARM1176JZF-S) Broadcom CPU. This is armv6k architecture, which is not compatible with recent builds of Ubuntu, which are optimized for performance at the expense of compatibility with older chip designs like this Broadcom's.
The obvious choice is to run the armel architecture of Debian -- you will feel almost at home as if you were on Ubuntu.
Revisiting this a few years later, there are now many models of R.PI available on the market:
Raspberry PI A, B, A+, B+, Zero, and Zero W: These are not compatible with Ubuntu ARM builds, as they use the BCM2835 Broadcom CPU (ARM v6k).
Raspberry PI 2, model B: this runs a BCM2836 Broadcom CPU. This is ARMv7 architecture, a Cortex A7 to be precise. This board (and presumably all subsequent "series 2" R.PI boards) will run Ubuntu just fine.
Raspberry PI 3, model A+, B, B+: these run on a BCM2837 Broadcom CPU. This is ARMv8 architecture, a Cortex-A53 64-bit chip to be exact. These board are a superior option over any R.PI 2 boards when it comes to running Ubuntu. Revision 1.2 of the Raspberry PI 2 uses this same 64 bit core, making the newer PI 2 boards significantly better than the original version.
Raspberry PI 4, model B: this one is based on the BCM2711, a Quad core Cortex-A72 (ARM v8) 64-bit chip almost thrice as fast as the SoC in PI model 3 boards. This board is now the best option to run Ubuntu on Raspberry PI.
Canonical has released Ubuntu Core (aka "Snappy") for the R.PI 2, but it bears noting this is a server distribution with no desktop environment. Additionally, standard Ubuntu 16.04 and 18.04 server images that can be expanded to include a desktop environment are available for R.PI 2 and R.PI 3. Ubuntu MATE is now also available for Raspberry PI.
Finally, starting with 20.04, official Ubuntu Desktop images are available for Raspberry PI. As long as Raspbian continues to ship a 32-bit userland, this should be considered the best fully 64-bit option for Raspberry Pi.
All current Raspberry PI images are linked from this page on the Ubuntu wiki.