How can I align YouTube embedded video in the center in bootstrap

You dont have to put <iframe> in a parent div at all. You can target exactly youtube iframe with CSS/3:

iframe[src*="//youtube.com/"], iframe[src*="//www.youtube.com/"] {
   display: block;
   margin: 0 auto;
}

Youtube uses iframe. You can simply set it to:

iframe {
   display: block;
   margin: 0 auto;
}

<iframe style="display: block; margin: auto;" width="560" height="315" src="https://www.youtube.com/embed/ig3qHRVZRvM" frameborder="0" allowfullscreen></iframe>

An important thing to note / "Bootstrap" is just a bunch of CSS rules

a fiddle

HTML

<div class="your-centered-div">
    <img src="http://placehold.it/1120x630&text=Pretend Video 560x315" alt="" />
</div>

CSS

/* key stuff */
.your-centered-div {
    width: 560px; /* you have to have a size or this method doesn't work */
    height: 315px; /* think about making these max-width instead - might give you some more responsiveness */

    position: absolute; /* positions out of the flow, but according to the nearest parent */
    top: 0; right: 0; /* confuse it i guess */
    bottom: 0; left: 0;
    margin: auto; /* make em equal */
}

Fully working jsFiddle is here.

EDIT

I mostly use this these days:

straight CSS

.centered-thing {
    position: absolute;
    margin: auto;
    top: 50%;
    left: 50%;
    transform: translate(-50%, -50%);
}

if your use stylus/mixins ( you should... it's the best )

center-center()
    absolute()
    margin auto
    top 50%
    left 50%
    transform translate(-50%,-50%)

This way... you don't need to know the size of the element - and the translate is based of it's size - So, -50% of itself. Neat.