Tomcat is a servlet container, and the following is how a servlet container works

The init, service, and destroy methods are the servlet's lifecycle methods. The init method is called once by the servlet container after the servlet class has been instantiated to indicate to the servlet that it being placed into service. The init method must complete successfully before the servlet can receive any requests. A servlet programmer can override this method to write initialization code that needs to run only once, such as loading a database driver, initializing values, and so on. In other cases, this method is normally left blank.

The service method is then called by the servlet container to allow the servlet to respond to a request. The servlet container passes a javax.servlet.ServletRequest object and a javax.servlet.ServletResponse object. The ServletRequest object contains the client's HTTP request information and the ServletResponse encapsulates the servlet's response. These two objects enable you to write custom code that determines how the servlet services the client request.

The servlet container calls the destroy method before removing a servlet instance from service. This normally happens when the servlet container is shut down or when the servlet container needs some free memory. This method is called only after all threads within the servlet's service method have exited or after a timeout period has passed. After the servlet container calls destroy, it will not call the service method again on this servlet. The destroy method gives the servlet an opportunity to clean up any resources that are being held (for example, memory, file handles, and threads) and make sure that any persistent state is synchronized with the servlet's current state in memory.

For Better Picture, Visit below article full of images to clear the concept :

http://shivasoft.in/blog/java/servlet/how-container-handles-the-servlet-request/

A Servlet Mapping is a directive in the web.xml that tells the Servlet Container which class to use when a particular Servlet is called. A Servlet is a class within your Java Web Application. Let's say you have a servlet called MyServlet in the com.example.servlet package. You would need to have a Servlet Mapping pointing the path "/MyServlet" to the "com.example.servlet.MyServlet" class. Without the servlet mapping, you would not be able to invoke your servlet because the Servlet container would not know where it is. JSPs are different - they do not need mappings like this. JSPs exist within the WebRoot of the application, so they are always available. Servlets exist in the WEB-INF\Classes directory once your application is deployed.

In send Redirect whenever the client makes any request it goes to the container, there the container decides whether the concerned servlet can handle the request or not. If not then the servlet decides that the request can be handle by other servlet or jsp. Then the servlet calls the sendRedirect() method of the response object and sends back the response to the browser along with the status code. Then the browser sees the status code and look for that servlet which can now handle the request. Again the browser makes a new request, but with the name of that servlet which can now handle the request and the result will be displayed to you by the browser. In all this process the client is unaware of the processing.

Because it is not a regular Java Class that is executed by a JVM. It is a special Java class that is executed by the Web Container which initializes and loads the servlet and the service methods get executed everytime it receives a request.

ServletRequest provides the functionalities to the classes to access the HTTP requests. The servlet container creates an HttpServletRequest object and an HttpServletResponse object, and passes them as arguments to the servlet's service methods such as doGet, doPost, etc.