@magi182's solution is solid, but it lacks the ability to set a maximum width. I think a maximum width of 640px is necessary because otherwhise the youtube thumbnail looks pixelated.
My solution with two wrappers works like a charm for me:
.videoWrapperOuter {
max-width:640px;
margin-left:auto;
margin-right:auto;
}
.videoWrapperInner {
float: none;
clear: both;
width: 100%;
position: relative;
padding-bottom: 50%;
padding-top: 25px;
height: 0;
}
.videoWrapperInner iframe {
position: absolute;
top: 0;
left: 0;
width: 100%;
height: 100%;
}
<div class="videoWrapperOuter">
<div class="videoWrapperInner">
<iframe src="//www.youtube.com/embed/C6-TWRn0k4I"
frameborder="0" allowfullscreen></iframe>
</div>
</div>
I also set the padding-bottom in the inner wrapper to 50 %, because with @magi182's 56 %, a black bar on top and bottom appeared.