When installing Ubuntu Server 14.04, you have the ability to automatically install and configure a LAMP stack. However, doing so in the latest build (as of 08/01/2015), seems to cause an ownership issue on a few directories that will prevent you from being able to actually start the service.
To fix this, you simply need to change the ownership of three directories using the following commands:
sudo chown -R mysql:mysql /var/run/mysqld sudo chown -R mysql:mysql /var/lib/mysql sudo chown -R mysql:mysql /usr/share/mysql
If you have changed the user that MySQL runs as, it goes without saying you will have to change the username in the above commands.
After setting the correct ownership on the directories, startup MySQL again using:
sudo service mysql start