Ville1955,
MySQL is available in a clean and easy to use installer. We are assuming you are on Windows. If this is not the case, please let us know what system you're having.
You can download it directly from
http://dev.mysql.com/downloads/mysql/ . The Web site will detect your Windows version and will offer the best MySQL match for download. Click on the "Download" button in the "Recommended Download:" section.
Run the installer and follow the on screen instructions. Always validate the default values (they are good for most uses).
At some stage of the installation you will be asked to choose a password for the database. This password is for the database administrator (root) and will be useful for accessing MySQL and configuring scripts (such as PhpMyAdmin). So note it down and do not forget it.
To manage the SQL database, you can install later a Web application/script such as the popular PhpMyAdmin.
Our recommendation though is to start with a free Windows tool to manage MySQL databases (similar to PhpMyAdmin but it does not require any setup on the web server). It is called SQLyog and is available from
https://github.com/webyog/sqlyog-community/wiki/Downloads .
By the way, the "connection" between MySQL and PHP is done by correctly configuring the Web application (such as Wordpress) and letting them know where MySQL is installed (locally or on a remote server) and giving them your MySQL username/password.
The Web server has not to know (and does not care) about MySQL. It requires that PHP is well declared only. And it is up to PHP to run your Web applications code, and it is that code which instructs PHP or its libraries to connect to the MySQL database and get/update data from it.