Update 2016:

Modern browser behave much better. All you should need to do is to set the image width to 100% (demo)

.container img {
   width: 100%;
}

Since you don't know the aspect ratio, you'll have to use some scripting. Here is how I would do it with jQuery (demo):

CSS

.container {
    width: 40%;
    height: 40%;
    background: #444;
    margin: 0 auto;
}
.container img.wide {
    max-width: 100%;
    max-height: 100%;
    height: auto;
}
.container img.tall {
    max-height: 100%;
    max-width: 100%;
    width: auto;
}​

HTML

<div class="container">
 <img src="http://i48.tinypic.com/wrltuc.jpg" />
</div>
<br />
<br />
<div class="container">
 <img src="http://i47.tinypic.com/i1bek8.jpg" />
</div>

Script

$(window).load(function(){
 $('.container').find('img').each(function(){
  var imgClass = (this.width/this.height > 1) ? 'wide' : 'tall';
  $(this).addClass(imgClass);
 })
})

Update 2019.

You can now use the object-fit css property that accepts the following values: fill | contain | cover | none | scale-down

https://developer.mozilla.org/en-US/docs/Web/CSS/object-fit

As an example, you could have a container that holds an image:

<div class="container">
    <img src="" class="container_img" />
</div>

.container {
    height: 50px;
    width: 50%;
}

.container_img {
    height: 100%;
    width: 100%;
    object-fit: cover;
} 

That's impossible with just HTML and CSS, or at least wildly exotic and complicated. If you're willing to throw some javascript in, here's a solution using jQuery:

$(function() {
    $(window).resize(function() {
        var $i = $('img#image_to_resize');
        var $c = $img.parent();
        var i_ar = $i.width() / $i.height(), c_ar = $c.width() / $c.height();            
        $i.width(i_ar > c_ar ? $c.width() : $c.height() * (i_ar));
    });
    $(window).resize();
});

That will resize the image so that it will always fit inside the parent element, regardless of it's size. And as it's binded to the $(window).resize() event, when user resizes the window, the image will adjust.

This does not try to center the image in the container, that would be possible but I guess that's not what you're after.

If you want to set a max width or height (so that it will not be very large) while keeping the images aspect-ratio, you can do this:

img{
   object-fit: contain;
   max-height: 70px;
}

I had similar issue. I resolved it with just CSS.

Basically Object-fit: cover helps you achieve the task of maintaining the aspect ratio while positioning an image inside a div.

But the problem was Object-fit: cover was not working in IE and it was taking 100% width and 100% height and aspect ratio was distorted. In other words image zooming effect wasn't there which I was seeing in chrome.

The approach I took was to position the image inside the container with absolute and then place it right at the centre using the combination:

position: absolute;
top: 50%;
left: 50%;
transform: translate(-50%, -50%);

Once it is in the centre, I give to the image,

// For vertical blocks (i.e., where height is greater than width)
height: 100%;
width: auto;

// For Horizontal blocks (i.e., where width is greater than height)
height: auto;
width: 100%;

This makes the image get the effect of Object-fit:cover.

Here is a demonstration of the above logic.

https://jsfiddle.net/furqan_694/s3xLe1gp/

This logic works in all browsers.

Many of the solutions found here have some limitation: some not working in IE ( object-fit) or older browsers, other solutions do not scale up the images (only shrink it), many solution do not support resize of the window and many are not generic, either expect fix resolution or layout(portrait or landscape)

If using javascript and jquery is not a problem I have this solution based on the code of @Tatu Ulmanen. I fixed some issues, and added some code in case the image is loaded dinamically and not available at begining. Basically the idea is to have two different css rules and apply them when required: one when the limitation is the height, so we need to show black bars at the sides, and othe css rule when the limitation is the width, so we need to show black bars at the top/bottom.

function applyResizeCSS(){
    var $i = $('img#imageToResize');
    var $c = $i.parent();
    var i_ar = Oriwidth / Oriheight, c_ar = $c.width() / $c.height();  
    if(i_ar > c_ar){
        $i.css( "width","100%");
        $i.css( "height","auto");          
    }else{
        $i.css( "height","100%");
        $i.css( "width","auto");
    }
}   
var Oriwidth,Oriheight;
$(function() {
    $(window).resize(function() {
        applyResizeCSS();
    });

    $("#slide").load(function(){
        Oriwidth  = this.width,
        Oriheight = this.height; 
        applyResizeCSS();
    }); 

    $(window).resize();
}); 

For an HTML element like:

<img src="images/loading.gif" name="imageToResize" id="imageToResize"/> 

try this

HTML:

<div class="container"></div>

CSS:

.container{
background-image: url("...");
background-size: 100%;
background-position: center;
}