LoginWithServlet
   Web Pages
      WEB-INF
         web.xml
      login.xhtml
      registr.xhtml
      index.xhtml
   Source Packages
      classes
         User.java
         Database.java
      filters
         LoginFilter.java
      servlets
         LoginServlet.java
         RegistrServlet.java

@WebServlet(name = "LoginServlet", urlPatterns = {"/login"})
public class LoginServlet extends HttpServlet {

    private static final Logger logger = Logger.getLogger(LoginServlet.class.getName());

    @Override protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException {
        request.getRequestDispatcher(request.getContextPath() + "/login.xhtml").forward(request, response);
    }

    @Override protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException {
        ServletContext ctx = request.getServletContext();

        String login = (String) request.getParameter("login");
        String password = (String) request.getParameter("password");

        Database database = Database.getInstance();
        User user = database.find(login, password);
        if (user == null) {
//            request.getRequestDispatcher(request.getContextPath() + "/registr").forward(request, response);
            ctx.log("Incorrect login or password");
            response.sendRedirect(request.getContextPath() + "/registr");
        }
        else {
            ctx.log(String.format("User %s has logged in", user));
            request.getSession().setAttribute("user", user);
            request.getRequestDispatcher(request.getContextPath() + "/index.xhtml").forward(request, response);
        }
    }
}

@WebServlet(name = "RegistrServlet", urlPatterns = {"/registr"})
public class RegistrServlet extends HttpServlet {

    private static final Logger logger = Logger.getLogger(RegistrServlet.class.getName());

    @Override protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException {
        request.getRequestDispatcher(request.getContextPath() + "/registr.xhtml").forward(request, response);
    }

    @Override protected void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException {
        ServletContext ctx = request.getServletContext();

        Long id = Long.valueOf(request.getParameter("id"));
        String login = (String) request.getParameter("login");
        String password = (String) request.getParameter("password");

        User user = new User(id, login, password);
        Database database = Database.getInstance();
        boolean inserted = database.insert(user);
        if(inserted) {
            ctx.log(String.format("New user %s has registred", user));
            request.getSession().setAttribute("user", user);
            request.getRequestDispatcher(request.getContextPath() + "/index.xhtml").forward(request, response);
        }
        else {
            ctx.log("Such user already exists");
        }
    }

    @Override public String getServletInfo() {
        return "Short description";
    }

}

@WebFilter("/*")
public class LoginFilter implements Filter {

    @Override public void doFilter(ServletRequest request,
                                   ServletResponse response,
                                   FilterChain chain) throws IOException, ServletException {
        HttpServletRequest req = (HttpServletRequest) request;
        HttpServletResponse res = (HttpServletResponse) response;
        HttpSession session = req.getSession(false);
        String loginURI = req.getContextPath() + "/login";

        boolean loggedIn = session != null && session.getAttribute("user") != null;
        boolean loginRequest = req.getRequestURI().equals(loginURI);

        if(loggedIn || loginRequest) {
            chain.doFilter(request, response);
        }
        else {
            res.sendRedirect(loginURI);
        }
    }
}