For some reason Grub is not finding the boot partition where it is supposed to be, without more information I could not tell you why. Possible causes are: An error when updating, a corrupted Grub bootloader, or a hardware error such as a failing hard drive. I'm sure there are others but these come to mind.
Most of the time fixing grub will solve these problems.
The easiest way to fix your grub is to use Boot-Repair
The detailed instructions and more info is HERE but the quick instructions are:
Boot into your LiveCD/USB and type the following into a terminal:
sudo add-apt-repository ppa:yannubuntu/boot-repair && sudo apt-get update
sudo apt-get install -y boot-repair && boot-repair
This will install and then run
boot-repair in your live environment(nothing will be installed to your hard drive)
When the window pops up click Recommended Repair this will scan your system for any OS's and then install grub with the correct information on the location of your Ubuntu partition and when you exit the live environment and reboot you should be able to load Ubuntu.
boot-repair runs it will create a file that will be uploaded to paste.ubuntu.com note the url that it gives you and you will be able to link to it to get more help. More information on this can be found on the link I gave above and well as more detailed instructions and information on advanced options(advanced options are usually not needed in a normal install)
If you want to check the hard drive the easiest way is with
Disk Utility which you can find in the dash by starting to type the name. When the first window pops up click on the icon that represents your harddisk on the left and you will be presented with the following screenshot. Click on the if the light is green where I have my mouse pointer yo should be good but if you want to test the surface of the disk you can do so by clicking that button and running the test on the next screen. These test will be long, an hour for a large disk would not be a surprise.