MySQL is an open source SQL relational database management system developed and supported by Oracle.
This is the short, one-sentence answer to the question “what is MySQL”, but let’s break it down into a little more human terms.
A database is just a structured collection of data that is organized for ease of use and retrieval. For a WordPress site, this “data” is things like the text of your blog posts, information for all registered users on your site, automatically loaded data, important settings configurations, and so on.
MySQL is just one popular system that can store and manage this data for you, and it’s a particularly popular database solution for WordPress sites.
What is a relational database?
A relational database is a collection of data items with predefined relationships between them. These items are organized into a set of tables consisting of columns and rows. Tables are used to list information about the objects that should be represented in the database. Each column in a table has some type of data, and each field indicates the actual value of an attribute. Rows in a table represent a set of related values of an object or entity. Each row in a table can be marked with a unique identifier, called a primary key, and rows from in multiple tables can be associated using foreign keys. This data can be accessed in several ways without the need to reorganize the database tables themselves.
Is Mysql still the right choice for a reliable relational database?
No, this is ALWAYS not the right choice.
MySQL and its MariaDB fork may be relatively reliable, but they are DBMSs that are extremely fast and efficient only if you are working on relatively small databases (a few hundred million records per table max). For larger databases, or requiring a particular point on transaction security, I would rather go for PostGRE, see SQL Server or Oracle (if you have a lot of money to invest), which are known for be extremely robust DBMS.
You should also keep in mind that MySQL / MariaDB is a lot of do it yourself, and you cannot afford to use these databases in production while keeping all the default configuration without even having see if there is anything to change. Reliability / security level, it can be a fortress as well as a colander if it gets put in the wrong hands.