Relational Databases
Relational databases are the most well known.
They store data in tables. These tables have rows (records) and columns (fields).
Just like a spreadsheet.
Different tables can have relationships - hence the name.
These relationships are usually created by connecting tables based on a common field.
For example, a pet owners database might have two tables - Pets & Owners.
To relate a pet to its owner, the Owner table might have an ID column. The Pet table might have an Owner ID column. The connection is made by saying - the Owner ID column in the Pet table connects to the ID column in the Owner table.
Relational databases are great for data that is neat and structured.
They power a large majority of most software.
One of the challenges with relational databases is that they need planning. You need a pretty good idea of what your data is going to look like before you build the database.
Other database models have popped up to address this.