My problem… getting an embedded YouTube video to automatically resize according to container sizes.

This is how you would post the embeded video:

<div class="videoWrapper">
    <!-- Copy & Pasted from YouTube -->
    <iframe width="560" height="349" src="" frameborder="0" allowfullscreen></iframe>

Then add the following custom CSS:

.videoWrapper {
	position: relative;
	padding-bottom: 56.25%; /* 16:9 */
	padding-top: 25px;
	height: 0;
.videoWrapper iframe {
	position: absolute;
	top: 0;
	left: 0;
	width: 100%;
	height: 100%;

Source with more info on other video sources: