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Can you elaborate how what you mean by connecting to web server... like what a browser does, or setting up a web server that supports JSP and servlets. For the "browser" thing, you'll need to use sockets, of which you can find more about from the Java API. For the server, you'll need a servlet compliant server, such as Tomcat. For more information on Tomcat, go to its website (tomcat.apache.com).
As for the session, it depends on where you need it. In a JSP, you can just use the implicit object session and call its getAttribute, setAttribute, and removeAttribute methods to modifiy session objects (notice I said objects, so primitive data types don't work). In a servlet, you'll need to first get the session by calling request.getSession() (and this is assuming your servlet is a subclass of HttpServlet, so the request is an HttpServletRequest).