The Ext License(s)
Ext uses a dual license model for the library. Basically, if you want to keep your source closed, you purchase a commercial license from Ext. If you keep your source open, you can use their open source option. The Ext site has a series of instructive pages describing license options.
Your options, in a nutshell, are described on the dual license page:
If you derive a commercial advantage by having a closed source solution, you must purchase an appropriate number of commercial licenses from Ext. By purchasing commercial licenses, you are no longer obligated to publish your source code.
If you wish to use the open source license of an Ext product, you must contribute all your source code to the open source community and you must give them the right to share it with everyone too.
If you plan to have an open source license on all code distributed with your application, then you can redistribute the Ext code with the GNU GPL license v3. Note that Ext provides an exception for open source applications using their code. This allows you to license the rest of your application with a different open source license than the GPLv3.
When you distribute an application that uses Ext, if you want to keep your source closed, then you must purchase a "commercial" license from Ext.
The GeoExt License
The GeoExt source may be distributed with a BSD license. Ext provides a license exception for open source frameworks using Ext that allows this license. Practically, this means you can modify and redistribute GeoExt without any special requirements. Keep in mind that the same is not true for Ext - if you redistribute the source or make any modifications, you are bound to the terms of their license.
If I create a web page using GeoExt and Ext, do I need a license on that page? If so, which licenses are applicable?
When you serve content on the web that includes source code, you are distributing that source code. By distributing the GeoExt source, you are bound to the terms of the GeoExt license. By distributing the Ext source, you are bound to the terms of the Ext license. Resources like images, CSS, and HTML are not covered by the GeoExt or Ext licenses. However, the Ext license may effect how you can license other source code that is associated with your web page. See the question below about distributing a application with Ext.
If I create an application using GeoExt and Ext and I want to distribute the code, can I? If so, how can I license my application?
The GeoExt license doesn't make any restrictions on what you can do when you distribute the GeoExt source. When you distribute the Ext source, you are bound the the terms of your Ext license.
If your application is an open source application and meets the terms on the following page, you can use any of the licenses described on the Ext page about licenses for open source applications.
If your application is closed source, to distribute the Ext source code, you must purchase a "commercial" Ext license.