When updating Ubuntu, old kernel versions are kept in /boot so that if you find that the latest one doesn’t work out, you can just boot into an older one. Unfortunately this also means that eventually your /boot partition will fill up if you never remove old kernels.
I usually have 100MiB /boot partitions on most of my Ubuntu systems so they fill up in a few months. There is plenty of documentation on the web for “/boot full ubuntu” though a while ago I found a script that easily removes all except your latest kernel. Here it is (run as root).
sudo su dpkg --get-selections|grep 'linux-image*'|awk\
'{print $1}'|egrep -v "linux-image-$(uname -r)|linux-image-generic"\
|while read n;do apt-get -y remove $n; done exit
[UPDATE]
You may see one or more of these errors when running the script:
The link /vmlinuz is a damaged link
Removing symbolic link vmlinuz you may need to re-run your boot loader [grub]The link /initrd.img is a damaged link
Removing symbolic link initrd.img you may need to re-run your boot loader [grub]
If like me your boot loader is grub (shown in the square brackets) then you can fix it by simply running:
sudo update-grub
See also this Q & A where someone suggests that Ubuntu is supposed to be cleaning up by itself, or that you can run sudo apt-get autoremove