Shims helps when adding semantic tags to older IE browsers, but there is a huge range of other new HTML5 features that having varying support on browsers. 

Polyfills are JavaScript code and/or browser plug-ins that can provide older or less featured browsers with API support.  The best polyfills will detect the whether the current browser has native support, and only adds the functionality if necessary.  The Douglas Crockford JSON2.js library is an example of this approach: if the browser already supports the JSON object, nothing changes.  If JSON is not available, the library adds a JSON property in the global object.

This approach provides some big benefits:

• It lets you add great new HTML5 features to your web sites sooner.
• It lets the developer focus on writing to the up-and-coming standard rather than proprietary APIs.
• Where most one-off legacy code fixes tends to break down over time, well done polyfills will stop executing over time (as customer browsers natively support the feature) meaning polyfill code may not need to be tested against new browsers since they will execute the native methods instead.

Your should also remember that Polyfills represent an entirely separate code path (and sometimes plug-in) that requires testing for support.  Also Polyfills tend to run on older browsers, which often have slower JavaScript performance.  As a result you might find that performance on older browsers is not comparable.

When looking for Polyfills you can start by checking the Modernizr GitHub wiki or the HTML5 Please site.

For an example of a polyfill consider a page that writes a few geometric shapes on a <canvas>

<script src="jquery-1.7.1.min.js"><script>
<script>
       $(document).ready(function () {
           drawCanvas();
       });

       function drawCanvas() {
          var context = $("canvas")[0].getContext('2d');

	  //background
          context.fillStyle = "#8B0000";
          context.fillRect(5, 5, 300, 100);

          // emptybox
          context.strokeStyle = "#B0C4DE";
          context.lineWidth = 4;
          context.strokeRect(20, 15, 80, 80);

	  // circle
          context.arc(160, 55, 40, 0, Math.PI * 2, false);
          context.fillStyle = "#4B0082";
          context.fill();
</script>

The result is a simple static canvas with a box & a circle:

…to enable this functionality on a pre-canvas browser we can find a polyfill.  A check on html5please.com references  FlashCanvas.  Pull down the zip and extract the files (flashcanvas.js, flash10canvas.swf, etc) to a directory on your site.  Then based on the documentation you need to add a single line to your original HTML file:

<!--[if lt IE 9]><script src="flashcanvas.js"></script><![endif]—>

…and you have canvas functionality!  The IE conditional comments ensure that the library is only loaded in browsers where it is useful, improving page load & processing time.

Like all Polyfills, you should test to verify the functionality matches your expectations across browsers you need to support.  For instance the Flash Canvas home page advertises 70% support of HTML5 Canvas spec tests.