From XmlWebApplicationContext javadoc:
By default, the configuration will be taken from \"/WEB-INF/applicationContext.xml\" for the root context, and
In a web application, the architecture is usually divided into layers like the popular MVC structure. So a web app comprises basically of a layer that handles the client requests i.e HTTPRequests and a layer that services those requests .
To summarize : classes that are meant to handle the Http requests i.e the controllers which are mapped to urls come under the test-servlet.xml. This is called as WebapplicationContext containing only the beans that are required mainly to handle the client requests.
Now the next part is the Service/Dao layer that comprises of your business logic. Beans that perform such logic are loaded under ApplicationContext object.
Now you may ask why have they separated these things in to files or two different objects.
Its because, an application have the same business logic that can be used by multiple clients working on different protocol. You may use the same service layers to handle RMI as well as HTTP calls. So Spring created a parent context that is started us as an ApplicationContext. And then if your applicationhandles web requests, you can create a dispathcher servlet which has its own Webapplicationcontext and initialized as a child of the parent context. So all the parent beans can be referenced in the child and can be overiden but not vice versa
"Spring context" = a Spring ApplicationContext.
"root context", in terms of a web application, means the main context that's loaded and used by the webapp. Typically, you'll start the root context with a ContextLoaderListener.
The root context isn't really a "kind" of context. It's just a role that a context plays. You have one root context in a webapp. Other contexts are not the root context. They're usually children of the root context.
A namespace here refers to the scope of an instance of Spring's DispatcherServlet. All it's saying right there is that if you name your servlet "test" in your web.xml, then by convention, Spring will look for a file named "test-servlet.xml" to use as that dispatcher's context. Incidentally, each context like this which is created for a dispatcher becomes a child of the root context.
Edit: To answer your new questions: