Your boot partition may contain older images of Linux kernals - they can be removed, and there is a procedure to do this outlined in What is the safest way to clean up /boot partition?

Update 2014-06-28

It looks as though you will need to increase the size of your /boot partition. This is not terribly fast, and of course if something goes wrong you will be glad you made a backup!

It may actually be / faster / easier / better all around / to reinstall Ubuntu.

You cannot resize partitions while you are using them, and Ubuntu is using them right now. So, you will need to boot from a live USB, and choose 'try Ubuntu'. When that is running, you open a terminal window and enter

sudo apt-get gparted

This will load the partition manager onto the USB. You can then run the partition manager to resize the partitions of you hard disk.

You want to make your /boot partition (/dev/sda1) larger - to do this you need to have some space available right next to it. To do that, you will probably need to shrink another partition, and possibly move partitions on the disk as well. It's the resize and move which take a long, long time.

I'm attaching two links to relevant posts made in the past - good luck!

http://www.howtogeek.com/114503/how-to-resize-your-ubuntu-partitions/

How to resize partitions?