When a JSP is requested for the first time or when the webapp starts up, the servlet container will compile it into a class extending HttpServlet and use it during the webapp's lifetime. You can find the generated source code in the server's work directory. In for example Tomcat, it's the /work directory. On a JSP request, the servlet container will execute the compiled JSP class and send the generated output (usually just HTML/CSS/JS) through the webserver over network to the client side, which in turn displays it in the web browser.
Servlet is an Java application programming interface (API) running on the server machine, which intercepts requests made by the client and generates/sends a response. A well known example is the HttpServlet which provides methods to hook on HTTP requests using the popular HTTP methods such as GET and POST. You can configure HttpServlets to listen on a certain HTTP URL pattern, which is configurable in web.xml, or more recently with Java EE 6, with @WebServlet annotation.
When a Servlet is first requested or during webapp startup, the servlet container will create an instance of it and keep it in memory during the webapp's lifetime. The same instance will be reused for every incoming request whose URL matches the servlet's URL pattern. You can access the request data by HttpServletRequest and handle the response by HttpServletResponse. Both objects are available as method arguments inside any of the overridden methods of HttpServlet, such as doGet() and doPost().
JSF (JavaServer Faces)
JSF is a component based MVC framework which is built on top of the Servlet API, and provides components via taglibs which can be used in JSP or any other Java based view technology such as Facelets. Facelets is much more suited to JSF than JSP. It namely provides great templating capabilities such as composite components, while JSP basically only offers the <jsp:include> for templating, so that you're forced to create custom components with raw Java code (which is a bit opaque and a lot of tedious work in JSF) when you want to replace a repeated group of components with a single component. Since JSF 2.0, JSP has been deprecated as view technology in favor of Facelets.
As being a MVC (Model-View-Controller) framework, JSF provides the FacesServlet as the sole request-response Controller. It takes all the standard and tedious HTTP request/response work from your hands, such as gathering user input, validating/converting them, putting them in model objects, invoking actions and rendering the response. This way you end up with basically a JSP or Facelets (XHTML) page for View and a Javabean class as Model. The JSF components are been used to bind the view with the model (such as your ASP.NET web control does) and the FacesServlet uses the JSF component tree to do all the work.