This component is part of Bootstrap, specifically the Navbar.
Essentially, to implement this feature, you should:
<nav class="navbar navbar-default" role="navigation">
<div class="container-fluid">
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#bs-example-navbar-collapse-1">
<span class="sr-only">Toggle navigation</span>
<div class="btn btn-primary">Menu</div>
</button>
<a class="navbar-brand" href="#">Brand</a>
</div>
<div class="collapse navbar-collapse" id="bs-example-navbar-collapse-1">
<ul class="nav navbar-nav">
<li class="active"><a href="#">Link</a></li>
<li><a href="#">Link</a></li>
</ul>
</div><!-- /.navbar-collapse -->
</div><!-- /.container-fluid -->
</nav>
You can then apply custom CSS styles to customize Bootstrap's default appearance.