There are many ways to describe a Geographic Information System. Here are three working definitions (from David A. Hastings, 1992, Geographic Information Systems: A Tool for Geoscience Analysis and Interpretation):
(The minimal definition): A GIS is a hardware/software system for the storage, management, and (with hardcopy or screen graphic) selective retrieval capabilities of georeferenced data. Definitions like this one are often used by vendors and users of vector-only GIS, whose objective is sophisticated management and output of cartographic data.
(A parallel definition): A GIS is a hardware/software system for managing and displaying spatial data. It is similar to a traditional Data Base Management System, where we now think in spatial rather than in tabular terms, and where the "report writer" now allows output of maps as well as of tables and numbers. Thus we can consider a GIS a "spatial DBMS" as opposed to traditional "tabular DBMSs." Few people use this definition, but it might help to explain GIS to a DBMS user.
(A more aggressive definition): A GIS is a system of hardware, software, and data that facilitates the development, enhancement, modeling, and display of multivariate (e.g. multilayered) spatially referenced data. It performs some analytical functions itself, and by its analysis, selective retrieval and display capabilities, helps the user to further analyze and interpret the data. Properly configured, the GIS can model (e.g. synthetically recreate) a feature or phenomenon as a function of other features or phenomena which may be related - where all features or phenomena are represented (characterized) by spatial and related tabular data. The analytical objectives described here are sometimes controversial - and often given lip service by cartographic GIS specialists who have not yet seen what can be accomplished scientifically by a select few GISs that go beyond cartographic approaches.
Another definition can be found at the University of Edinburgh.