The Linux package management system doesn't allow us to install two MySQL instances. Instead, we implement separation in the same MySQL instance by using separate databases with different usernames and access privileges for each database.
The first step is to ensure that MySQL is installed on your server. For Ubuntu, DigitalOcean has a fairly good tutorial: https://www.digitalocean.com/community/tutorials/how-to-install-mysql-on-ubuntu-14-04. While the Ubuntu version for that tutorial is old, the instructions are still accurate enough.
The MySQL server must support TCP connections from localhost. Edit the configuration file, /etc/mysql/my.cnf, to have the following line:
bind-address = 127.0.0.1
