A database is a storage mechanism which allows you to store structured or unstructured data for future keeping. You can think of it as a spreadsheet like excel, but there are a lot more advantages to using a dedicated database within an application, one being the speed and performance that you will get from querying the data as well as the large data set that is possible to be stored within it.
Data in the database can be structured or unstructured through SQL or NoSQL versions of database technologies, respectively. There are potentially hundreds of different kinds of database technologies that are used. Some main ones are MySQL SQL and NoSQL.
When developing applications, code will be required to make a connection to the database, and then you can query the data from the database or insert data or update and delete it as required. Behind every application is a database structure that maintains the data that you are interacting with.