Any organised collection of data may be considered a database. In this sense the word is synonymous with dataset. It is a collection of data stored according to a schema and manipulated according to the rules set out in one Data Modelling Facility. Sources: ODH and USOD

Another meaning relates to a software system for processing and managing data, including features to extend or update, transform and query the data. Examples are the open source PostgreSQL, and the proprietary Microsoft Access. Source: ODH