Go to Ubuntu Software Center(Applications>Ubuntu Software Center)
Search for MySQL and find ‘MySQL Server’ in the list.
Install it. Use a password if you want one.
Open the Terminal(Shell or Command Line). Type the following to see if correctly installed.
mysql --version
For me, it returned the following
mysql Ver 14.14 Distrib 5.1.41, for debian-linux-gnu (i486) using readline 6.1
Once installed correctly, you have to start the mysql server. Use the following command.
sudo /etc/init.d/mysql start
If the root doesn’t have a password, use the following command to set password
mysqladmin -u root password myPassword
Then, to login to the server as root, use the following command
mysql -u root -p
Type in the password and just get ready for some awesomeness!

If you are a beginner, just read through this tutorial.
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