When you choose the default partition plan on a Ubuntu server you likely end up with a small boot partition.
If you don’t reboot your server on a regular basis you have a rather big chance of a full boot partition after a couple of automatic updates.
Option 1: you have a bit space left
When you apt-get is stell working normally you can do this to purge older kernels:
Show current running kernel: uname -r
Show installed kernels dpkg --list ‘linux-image*’ | grep \^ii
Delete all kernels excepts for the running kernel and the kernel with the highest number:
Purging a kernel: apt-get purge linux-image-3.13.0-51-generic
After that you can run a apt-get autoremove