These are some of our favorite client applications. There are some others out there and you can refer to the GeoServer client page for more information.

MapBuilder A javascript client that supports OGC WMS and WFS requests, so you can view and edit spatial data through the web browser. GeoServer has MapBuilder built in as its map preview tool.

Open Layers OpenLayers is a javascript tiling web-based client just like Google maps. It supports OGC WMS requests.

MapBender Mapbender is the software and portal site for geodata management of OGC OWS architectures. The software provides web technology for managing spatial data services implemented in PHP, JavaScript and XML. It provides a data model and interfaces for displaying, navigating and querying OGC compliant map services. The Mapbender framework furthermore provides authentication and authorization services, OWS proxy functionality, management interfaces for user, group and service administration in WebGIS projects.

Udig Udig is a desktop client that supports OGC WMS and WFS requests. It is built off of the same library as GeoServer: GeoTools.

GVSig GVSig is a desktop client that supports OGC WMS and WFS requests, similar to Udig.

Google Earth GeoServer can serve up data as KML/KMZ that can then be loaded in Google Earth. There is a tutorial here that you can read to start viewing your in this client.

NASA World Wind World Wind supports WMS requests so you can serve up your data with GeoServer to it. Read this tutorial for how to do this.