As a Geography graduate I feel compelled to chime in on this one. Cetin has given a good response. Though it impossible to simplify the scope of GIS I usually explain it like this just to give an idea of how it works:

1. Take what you know about how a relational database works and apply it to maps.

2. Not just ANY maps. Maps that that represent layers of thematic information - take the following as examples:
- Soil type
- Elevation
- Political boundaries...city, county, census, zoning, congressional district, you name it.
- Flood plain boundaries
- Roads infrastructure
- Water and sewer infrastructure
...I'll stop here.

3. Now imagine that you have to decide where a Wal-Mart (substitute something more substantive if you wish) will be located based on the following:
- You must identify a location of thirty acres to be within .75 miles of a major arterial highway that sees an average of over 200,000 vehicles daily. It must also be in a low property tax district that has a minimal surface slope but still has enough to support good drainage and is accessible to water and sewer facilities. It must out of all FEMA-known flood plains. The local population must be in excess of 50,000 within a 30-mile radius. The per-capita income of those residents must meet a certain level. The site must be zoned for retail however if necessary you should be able to include farm zones adjacent to commercial zones since they will likely be property that can be purchased from a retired farmer (planning on selling it to a subdivision developer anyway) and can be re-zoned with a little greasing of the local city council.

4. You have to have this information (all potential locations) plotted onto a map and ready for presentation in under 30 minutes.

In the past this process could take months. By working with datasets from various sources (and knowing how to line them all up) GIS allows decision makers to query line, point, and polygon-oriented data to make very complex analysis of criteria that, in any other venue, would not be provided in a single map.

It is also useful for studying changes in environmental data (using remote sensing satellite imagery), population shifts, demographics, and other temporal data. Remember the queries I mentioned earlier? They can be applied to layers of the SAME data taken at different times (months, years, whatever) and filtered to show only what has changed within a certain threshold.

Very powerful stuff. It's a world all unto itself.

Alan