A widget is a collection of stylesheets, javascripts and layouts that add additional functionality to a slide deck. Let us take the example of mathjax, a widget that comes pre-bundled with slidify, that allows mathematical equations to be rendered beautifully.
Shown below are the contents of the mathjax widget folder.

mathjax.html is a mustache layout that specifies how this widget would be inserted into a slide deck. Note that it serves mathjax from a CDN when you are connected to the internet, and if not, it switches to a stripped down local version. The mustache variable url.widgets, specifies the path to the location of the widgets folder in the root directory, so that it is served correctly.
<!-- MathJax: Fall back to local if CDN offline but local image fonts are not supported (saves >100MB) -->
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
tex2jax: {
inlineMath: [['$','$'], ['\\(','\\)']],
processEscapes: true
}
});
</script>
<script src="https://c328740.ssl.cf1.rackcdn.com/mathjax/2.0-latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<script>
window.MathJax || document.write('<script type="text/x-mathjax-config">MathJax.Hub.Config({"HTML-CSS":{imageFont:null}});<\/script><script src="{{url.widgets}}/mathjax/MathJax.js?config=TeX-AMS-MML_HTMLorMML"><\/script>')
</script>
Slidify automatically inserts the contents of a layout file if it has the same name as that of the widget.