MySQL is a database management system that is used by WordPress to store and retrieve all your blog information. Think of it this way. If your database is a filing cabinet that WordPress uses to organize and store all the important data from your website (posts, pages, images, etc), then MySQL is the company that created this special type of filing cabinet.
MySQL is an open source relational database management system. It runs as a server and allows multiple users to manage and create numerous databases. It is a central component in the LAMP stack of open source web application software that is used to create websites. LAMP stands for Linux, Apache, MySQL, and PHP. Most WordPress installations use the LAMP stack because it is open source and works seamlessly with WordPress.
WordPress requires MySQL to store and retrieve all of its data including post content, user profiles, and custom post types. Most web hosting providers already have MySQL installed on their web servers as it is widely used in many open source web applications such as WordPress.
WordPress uses the PHP programming language to store and retrieve data from the MySQL database. To retrieve data from the database, WordPress runs SQL queries to dynamically generate content. SQL stands for Structured Query Language and is the programming language typically used to query databases.
For users that are not comfortable writing their own PHP and SQL scripts, most web hosting providers offer easy to use web applications to manage databases. One such web application is phpMyAdmin which allows users to manage their database using a web based graphical interface. You can manipulate your tables visually while phpMyAdmin runs the SQL queries for you.