A database is a computer system for storing and taking care of data (any kind of information).
When the software that runs a database is separate from the programs that use the database, it is called a database engine.
Information stored inside a database is usually stored in an organized way. Data about a person that would have been written on a piece of paper before databases would be stored in a record in a database. A collection of person records that would have been an address book before databases would be stored in a file or table.