A database is an organised collection of data. There is usually an implication that the data is stored in and managed by a computer or collection of computers, in which case the definition should perhaps be:
A database is an organised collection of data that is managed by a DBMS (Database Management System) (q.v.)
Often the word database is used when what is really meant is a DBMS.
There is a tendency to see a database (or DBMS) as merely a system for the persistent storage and retrieval of data, but this view tends to miss the organised part of the definition.