To make it work, you have to place the alert() in the success function:
$.ajax({
type: "GET",
url: "Testpage.aspx",
data: "name=test",
dataType: "html",
success: function(mydata) {
alert(mydata);
}
});
AJAX calls are asynchronous, and therefore JavaScript would evaluate alert(v_name); before the server responds to the AJAX call, and therefore before the success function is called.
Your AJAX applications must be designed in such a way to be driven by the AJAX response. Therefore anything you plan to do with mydata should be invoked from the success function. As a rule of the thumb, imagine that the server will take very long (such as 1 minute) to respond to the AJAX request. Your program logic should work around this concept of asynchrony.