How to post and access the data using jquery ajax in asp.net webforms?

This looks like you want to use WebMethod in ASP.NET:

$(document).ready(function () {
            $.ajax({
                url: 'Default2.aspx/HelloWorld',
                data: "{ 'name': '" + "randel" + "' }",
                type: "POST",
                contentType: "application/json; charset=utf-8",
                dataType: "json",
                success: function (data) {
                    // alert('insert was performed.');
                    $("#mydiv").empty();
                    $("#mydiv").html(data);
                },
                error: function (data, status, jqXHR) { alert(jqXHR); }
            });
        });

and in your code behind you should do this:

[WebMethod()]
public static string HelloWorld(string name)
{
string message = "Hello " + name;
return message;
}

WebMethods are in some kind better than doing __doPostBack() because you controll all client-server traffic using for example jQuery. More about WebMethods: here or just google WebMethods ASP.NET.

And If you want to receive some form value you should put it on $.ajax data parameter and add the same parameter in WebMethod.

EDITED

From the code you post I see that you want send from Default.aspx some data to Default2.aspx and load some content from Default2.aspx (#div).

You can do this:

$.ajax({
                url: "/Default2.aspx",
                type: "GET",
                dataType: "html",
                async: false,
                data: { "name": "randel"
                },
                success: function (obj) {
                    // obj will contain the complete contents of the page requested
                    // use jquery to extract just the html inside the body tag
                    $content = $(obj).find('body #div').html();
                    // then update the dialog contents with this and show it
                }
         });

And in code behind:

protected void Page_Load(object sender, EventArgs e)
    {
        Label1.Text = Request.QueryString["name"];
    }