We describe the application of data mining algorithms to research problems in astronomy. We posit that data mining has always been fundamental to astronomical research, since data mining is the basis of evidence-based discovery, including classification, clustering, and novelty discovery. These algorithms represent a major set of computational tools for discovery in large databases, which will be increasingly essential in the era of data-intensive astronomy. Historical examples of data mining in astronomy are reviewed, followed by a discussion of one of the largest data-producing projects anticipated for the coming decade: the Large Synoptic Survey Telescope (LSST). To facilitate data-driven discoveries in astronomy, we envision a new data-oriented research paradigm for astronomy and astrophysics -- astroinformatics. Astroinformatics is described as both a research approach and an educational imperative for modern data-intensive astronomy. An important application area for large time-domain sky surveys (such as LSST) is the rapid identification, characterization, and classification of real-time sky events (including moving objects, photometrically variable objects, and the appearance of transients). We describe one possible implementation of a classification broker for such events, which incorporates several astroinformatics techniques: user annotation, semantic tagging, metadata markup, heterogeneous data integration, and distributed data mining. Examples of these types of collaborative classification and discovery approaches within other science disciplines are presented.