i hope this concise explanation will give you an overview and the understanding you expect.
PART A
SERVER SIDE
In your web server application on your server, if using Java, you are to create a Java servlet class to process data that was submitted from client browser via script or form and to provide dynamic content such as the results of a database query from the client.
Read more on Servlets from:
- http://docs.oracle.com/javaee/5/tutorial/doc/bnafe.html
- http://en.wikipedia.org/wiki/Java_Servlet
- What is Java Servlet?
Also read more about how to register your servlet on the server (web.xml for java Projects)
Example of a servlet:
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@WebServlet(name = "MyServlet", urlPatterns = {"/calculator"}, asyncSupported = true)
public class MyServlet extends HttpServlet {
@Override
public void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException {
    Enumeration e = request.getParameterNames(); // parsing the string from client
    while (e.hasMoreElements()) {
        String name = (String) e.nextElement();// eg. "command" from ajax
        String value = request.getParameter(name); // eg. getSum
        if (value.equals("getSum")) {
            // Instantiate a java class and call the method
            // that performs the addition and returns the value
            Calculator calc = new Calculator();
            String answer = (String) calc.getSum();
            if (answer != null) {
                // Set contentType of response to client or browser
                // so that jQuery knows what to expect.
                response.setContentType("text/plain");
                PrintWriter out = response.getWriter();
                // return answer to ajax calling method in browser
                out.print(answer); 
                out.close();
            }
        }
    } // END While LOOP
}
@Override
public void doPost(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException {
    // include method if you call POST in you ajax on client side
}
}
a Java Class for calculations on your server path
public class Calculator {
    public int getSum() {
        return 10+15;
    }
}
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PART B
CLIENT SIDE – Your Browser
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You have to visit jQuery website, download and add the jQuery ajax script to your project. “jquery-ui.min.js” is sufficient for this purpose. Add this script to your html or jsp file using the following line:
 <script src="resources/ajax/libs/jqueryui/1.8/jquery-ui.min.js" type="text/javascript"></script>
Within your external javascript file or inline javascript include  a function to call the servlet and get the sum as follows: 
function getSum(){
    $.ajax({
        type: 'GET',       //  type of request to make. method doGet of the Servlet will execute
        dataType: 'text',  // specifying the type of data you're expecting back from the server
        url: 'calculator', // the URL to send the request to. see annotation before class declaration
        data: "command="+"getSum", // Data to be sent to the server (query string)
        // if request fails this method executes
        error: 
        function(e){
            alert('Error. Unable to get response from server');
        },
        // when request is successful, this function executes
        // display the data from server in an alert
        success: 
            function(result){
                if(result) {
                    alert(result);
                }
            }    
   });
}