Create user and db in MySQL for various apps

Working on a portal which requires the integration of various opensource applications, I had to install quite a few systems that require mysql databases. In order to keep a safe separation between these systems, one should create different users that will have access only to the specific database. This is a simple way to prevent losing all your databases in case of  the mishap when an attack is successful on a system and sql commands pass directly to the database (sql injection etc). This is a how-to create the new user, the new database and set the requested permissions.