The broadest definition for the term database is any collection of information that is organized according to some principle or rule. A telephone book is a database in this sense. It is organized alphabetically. The Library's main book collection is a database in this sense, too. It is organized according to the Library of Congress Subject Classification System .
Organization is the key characteristic of a database.
It could be in a computer, but it wouldn't have to be. It could be on 3 X 5 cards and still be a database. If my recipe collection is a pile of cards thrown randomly into a shoebox, though, it is not a database, even though it might have the same recipes as the organized collection. The World Wide Web is generally considered not to be a database because it is not organized. In contrast, the Yahoo subject directory is a database, organized by subject.