Download the image and then:
Linux
Extract the .img.bz2 archive to get the image file.
bunzip2 ubuntu-mate-15.04-desktop-armhf-banana-pro.img.bz2
Write the image file to the microSD card as root.
sudo ddrescue -d -D --force ubuntu-mate-15.04-desktop-armhf-banana-pro.img /dev/sdX
download link: ubuntu-mate-15.04-desktop-armhf-banana-pro.img.7z
login: root
password: bananapi
Windows
Windows users might want to use Win32 Disk Imager
Download: http://sourceforge.net/projects/win32diskimager/
Changes:
Added xserver-xorg-video-fbturbo (an accelerated x.org driver)
Added kernel linux-bananapi (https://github.com/Bananian/linux-bananapi)