A DBMS 1) is an application that controls:
A database is a collection of permanently stored data used by an application or enterprise. A database contains logically related data, which means that the database was created to represent the real-world.
All databases are not built on the relational model, see Database - Model
When people use the word database, fundamentally what they are saying is that the data should be self-describing and it should have a schema. That's really all the word database means.
The databases were born to insert into.
A collection of information organized to afford efficient retrieval
They are also known as storage engine.
After writing data:
Why would I want a database? What problem do they solve?