MySQL is a Relational Database Management System. A relational database adds speed and flexibility, by storing data in separate tables rather than putting all the data in one area. These tables are linked by defined relations making it possible to combine data from several tables upon request.
Using a RDMS means it is possible to add, access, and process the data stored in your database.
'SQL' stands for "Structured Query Language" - the most common standardized language used to access databases.