What is GeoServer?
GeoServer, as the name suggests, is a server. It is open source and it allows users to view and edit geographic data. This is the realm of Geographic Information Systems (GIS). GeoServer is a fully functional WFS-T and WMS server that follows the OGC specifications.
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With Geoserver you can publish data as:
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In short, GeoServer is one of those tools you need to display maps on web pages, where the user can zoom and pan around. It is used in conjunction with clients such as, OpenLayers (for web pages), Google Earth, UDig, GVSig and others. The use of standards allows information from GeoServer to be easily combined with other geographic information.

GeoServer's focus is ease of use and support for open standards, in order to enable anyone to quickly share their geospatial information in an interoperable way. More information on specific features of GeoServer can be found here.
Here is an example of what you might get if you make a WMS request to GeoServer:

Geoserver is built on Geotools, the same Java toolkit that UDig uses.
It is open source, and the license agreement is here.