Above of all, I found this library right here. Works out of the box perfectly on a Bootstrap-3 environment.

Bootstrap-3 Clock-Picker

CSS

<link rel="stylesheet" type="text/css" href="dist/bootstrap-clockpicker.min.css">

HTML

<div class="input-group clockpicker">
    <input type="text" class="form-control" value="09:30">
    <span class="input-group-addon">
        <span class="glyphicon glyphicon-time"></span>
    </span>
</div>

JAVASCRIPT

<script type="text/javascript" src="dist/bootstrap-clockpicker.min.js"></script>
<script type="text/javascript">
    $('.clockpicker').clockpicker();
</script>

As simple as that! Find more examples on the link above.

Update 18/04/2018

If you are using Bootstrap-4, the most popular time/date picker library available right now is Tempus Dominus. It is not fancy looking, but much responsive and modern.

Bootstrap-4 Tempus Dominus