noun A collection of (usually) organized information in a regular structure, usually but not necessarily in a machine-readable format accessible by a computer.
I have a database of all my contacts in my personal organizer.
noun A set of tables and other objects (queries, reports, forms) in the form of a structured data set.
The "books" database will have three tables, and the "customers" database will have two tables.
noun A software program (application) for storing, retrieving and manipulating such a structured data set.
Which database do you use: MySQL or Oracle?
noun A combination of such data sets and the programs for using them.