How can I obtain the local time in jQuery?

You can get the local timezone offset of the client to get the GMT time and then add the offset hours of the Germany timezone (Central European Time GMT+1):

function getDate(offset){
  var now = new Date();
  var hour = 60*60*1000;
  var min = 60*1000;
  return new Date(now.getTime() + (now.getTimezoneOffset() * min) + (offset * hour));
}

//...

var dateCET = getDate(1); // Central European Time is GMT +1

if (dateCET.getHours() < 12) {
    alert ("Good morning.");
} else {
    alert ("Good afternoon.");
}

Update: I agree with @Josh, the above code is completely client dependent. Let’s try to do it better:

$(document).ready(function(){
  var timezone = "Europe/Berlin";
  $.getJSON("http://json-time.appspot.com/time.json?tz="+timezone+"&callback=?",
    function(data){
      if (data.hour < 12) {
        alert ("Good morning in "+timezone);
      } else {
        alert ("Good afternoon in "+timezone);
      }
    })
});

We are now taking advantage of JSONP to do Cross-Domain requests to the jsontime server, this server exposes a complete JSON API to query time and timezone information.

You can play with the code here and you can explore the JSONP API here.

Hurray!, no server-side code!