Usually when I see a Bootstrap component in the documentation, and I want to replicate it, I just do CTRL+U in Chrome (or whatever shortcut you have for source viewing in your browser) and copy the code and paste it in my favorite editor and then I have that component. In your case, the mark up would be:
<div class="container">
<nav class="navbar navbar-default">
<div class="container-fluid">
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar" aria-expanded="false" aria-controls="navbar">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="#">Project name</a>
</div>
<div id="navbar" class="navbar-collapse collapse">
<ul class="nav navbar-nav">
<li class="active"><a href="#">Home</a></li>
<li><a href="#">About</a></li>
<li><a href="#">Contact</a></li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-haspopup="true" aria-expanded="false">Dropdown <span class="caret"></span></a>
<ul class="dropdown-menu">
<li><a href="#">Action</a></li>
<li><a href="#">Another action</a></li>
<li><a href="#">Something else here</a></li>
<li role="separator" class="divider"></li>
<li class="dropdown-header">Nav header</li>
<li><a href="#">Separated link</a></li>
<li><a href="#">One more separated link</a></li>
</ul>
</li>
</ul>
<ul class="nav navbar-nav navbar-right">
<li class="active"><a href="./">Default <span class="sr-only">(current)</span></a></li>
<li><a href="../navbar-static-top/">Static top</a></li>
<li><a href="../navbar-fixed-top/">Fixed top</a></li>
</ul>
</div><!--/.nav-collapse -->
</div><!--/.container-fluid -->
</nav>
</div>
Note that the nav element is wrapped in a container element, that's what you don't want to be doing really. I'd rather just leave nav as a direct child of body.
JS Fiddle