How can I make a div stick to the top of the screen once it's been scrolled to?

You could use simply css, positioning your element as fixed:

.fixedElement {
    background-color: #c0c0c0;
    position:fixed;
    top:0;
    width:100%;
    z-index:100;
}

Edit: You should have the element with position absolute, once the scroll offset has reached the element, it should be changed to fixed, and the top position should be set to zero.

You can detect the top scroll offset of the document with the scrollTop function:

$(window).scroll(function(e){ 
  var $el = $('.fixedElement'); 
  var isPositionFixed = ($el.css('position') == 'fixed');
  if ($(this).scrollTop() > 200 && !isPositionFixed){ 
    $el.css({'position': 'fixed', 'top': '0px'}); 
  }
  if ($(this).scrollTop() < 200 && isPositionFixed){
    $el.css({'position': 'static', 'top': '0px'}); 
  } 
});

When the scroll offset reached 200, the element will stick to the top of the browser window, because is placed as fixed.

You've seen this example on Google Code's issue page and (only recently) on Stack Overflow's edit page.

CMS's answer doesn't revert the positioning when you scroll back up. Here's the shamelessly stolen code from Stack Overflow:

function moveScroller() {
    var $anchor = $("#scroller-anchor");
    var $scroller = $('#scroller');

    var move = function() {
        var st = $(window).scrollTop();
        var ot = $anchor.offset().top;
        if(st > ot) {
            $scroller.css({
                position: "fixed",
                top: "0px"
            });
        } else {
            $scroller.css({
                position: "relative",
                top: ""
            });
        }
    };
    $(window).scroll(move);
    move();
}

<div id="sidebar" style="width:270px;"> 
  <div id="scroller-anchor"></div> 
  <div id="scroller" style="margin-top:10px; width:270px"> 
    Scroller Scroller Scroller
  </div>
</div>

<script type="text/javascript"> 
  $(function() {
    moveScroller();
  });
</script> 

And a simple live demo.

A nascent, script-free alternative is position: sticky, which is supported in Chrome, Firefox, and Safari. See the article on HTML5Rocks and demo, and Mozilla docs.

As of January 2017 and the release of Chrome 56, most browsers in common use support the position: sticky property in CSS.

#thing_to_stick {
  position: sticky;
  top: 0px;
}

does the trick for me in Firefox and Chrome.

In Safari you still need to use position: -webkit-sticky.

Polyfills are available for Internet Explorer and Edge; https://github.com/wilddeer/stickyfill seems to be a good one.