There is a list of Web Clients for Geoserver on the Geoserver site (http://geoserver.org/display/GEOSDOC/Clients). Unfortunately, it is posted as no longer maintained. Does anyone have a current list of web clients?
It depends on which services you want to provide/consume using Geoserver (WMS, WFS/WFS-T, or WCS). Based on the services you want to use, you can look for suitable clients. OpenLayers is one of the most advanced tools and documentation is good.
Other clients include:
you might also consider openscales for flex/AIR based geoserver clients.
I would guess that the most popular tool used to build Web clients for GeoServer is OpenLayers
You may also want to look at MapFish
http://mapfish.org/ and GeoExt
You might one to check one of the newest arrivals on the scene - CartoView.