How do I fit an image (img) inside a div and keep the aspect ratio?

Using CSS only:

div > img {
  width: auto;
  height : auto;
  max-height: 100%;
  max-width: 100%;
}

You will need some JavaScript to prevent cropping if you don't know the dimension of the image at the time you're writing the css.

HTML & JavaScript

<div id="container">
    <img src="something.jpg" alt="" />
</div>

<script type="text/javascript">
(function() {

var img = document.getElementById('container').firstChild;
img.onload = function() {
    if(img.height > img.width) {
        img.height = '100%';
        img.width = 'auto';
    }
};

}());
</script>

CSS

#container {
   width: 48px;
   height: 48px;
}

#container img {
   width: 100%;
}

If you use a JavaScript Library you might want to take advantage of it.

Use max-height:100%; max-width:100%; for the image inside the div.

CSS Object-fit

Unfortunately max-width + max-height do not fully cover my task... So I have found another solution:

To save the Image ratio while scaling you also can use object-fit CSS3 propperty.

Useful article: Control image aspect ratios with CSS3

img {
    width: 100%; /* or any custom size */
    height: 100%; 
    object-fit: contain;
}

Bad news: IE not supported (Can I Use)