A database is an application that manages data and allows fast storage and retrieval of that data.
There are different types of database, including a modern NoSQL but the most popular meaning is a relational database.
A relational database stores data in tables where each row in the table holds the same sort of information. In the early 1970s, Ted Codd, an IBM researcher devised 12 laws of normalization. These apply to how the data is stored and relations between different tables.
SQL is a simplistic programming language that is used in relational databases.