The key to creating a responsive YouTube embed is with padding and a container element, which allows you to give it a fixed aspect ratio. You can also use this technique with most other iframe-based embeds, such as slideshows.
Here is what a typical YouTube embed code looks like, with fixed width and height:
<iframe width="560" height="315" src="//www.youtube.com/embed/yCOY82UdFrw"
It would be nice if we could just give it a 100% width, but it won’t work as the height remains fixed. What you need to do is wrap it in a container like so (note the class names and removal of the width and height):
frameborder="0" allowfullscreen class="video"></iframe>
And use the following CSS:
How this works: The container element is given a zero height and a percentage bottom padding. The percentage bottom padding is a percentage of the container width, so that gives it a fixed aspect ratio. But in order to get the iframe to show up inside the zero-height container, you need to make the container relative and the iframe absolute, positioned inside the div.