Databases:
Databases are where serious applications and servers hold all their data. A program that allows databases to be created and modified is called a Database Management System (DBMS).
The language to communicate with a DBMS is SQL: Structured Query Language. Most DBMses accept all SQL commands. Some have quirks for SQL commands. Some have non standard SQL commands. But for basic stuff, any DBMS will do.

There are a number of floss DBMSes:
PostgreSQL
MariaDB

You may have also heard of some proprietary ones:
Oracle
Microsoft SQL "sequel" Server

Inside a database you will find a number of tables (think spreadsheets, not furniture). Each table has some fields/columns that store different types of data, and a number of entries/rows which hold that data. Tables will also have one or more Primary Keys, which are used to find a unique row. Often the primary key will just be a number, like a customer ID.
