Mapping is simply showing the association between data and geographic locations on a map.
Cartography is the science and art of making maps.
GIS, or geographic information systems, is a collection of software, data, and people that supports mapping, analysis and decision making. GIS enables you to interact with data spatially in ways that tables, charts and static maps cannot achieve. For further reading, check out Penn State's open GIS textbook: The Nature of Geographic Information.
Here are some examples of what GIS can do:
GIS has applications in almost every field of study - explore the possibilities in this library guide! You can also get started with some self-paced, introductory lessons by clicking either of the maps above. Lynda.com provides a video module called Real-World GIS that provides a high-level overview of GIS data and applications.