How to make a floated div 100% height of its parent?

For the parent:

display: flex;

You should add some prefixes http://css-tricks.com/using-flexbox/

Edit: Only drawback is IE as usual, IE9 does not support flex. http://caniuse.com/flexbox

Edit 2: As @toddsby noted, align items is for parent, and its default value actually is stretch. If you want a different value for child, there is align-self property.

Edit 3: jsFiddle: https://jsfiddle.net/bv71tms5/2/

For #outer height to be based on its content, and have #inner base its height on that, make both elements absolutely positioned.

More details can be found in the spec for the css height property, but essentially, #inner must ignore #outer height if #outer's height is auto, unless #outer is positioned absolutely. Then #inner height will be 0, unless #inner itself is positioned absolutely.

<style>
    #outer {
        position:absolute; 
        height:auto; width:200px; 
        border: 1px solid red; 
    }
    #inner {
        position:absolute; 
        height:100%; 
        width:20px; 
        border: 1px solid black; 
    }
</style>

<div id='outer'>
    <div id='inner'>
    </div>
    text
</div>

However... By positioning #inner absolutely, a float setting will be ignored, so you will need to choose a width for #inner explicitly, and add padding in #outer to fake the text wrapping I suspect you want. For example, below, the padding of #outer is the width of #inner +3. Conveniently (as the whole point was to get #inner height to 100%) there's no need to wrap text beneath #inner, so this will look just like #inner is floated.

<style>
    #outer2{
        padding-left: 23px;
        position:absolute; 
        height:auto; 
        width:200px; 
        border: 1px solid red; 
    }
    #inner2{
        left:0;
        position:absolute; 
        height:100%; 
        width:20px; 
        border: 1px solid black; 
   }
</style>

<div id='outer2'>
    <div id='inner2'>
    </div>
    text
</div>

I deleted my previous answer, as it was based on too many wrong assumptions about your goal.

Actually, as long as the parent element is positioned, you can set the child's height to 100%. Namely, in case you don't want the parent to be absolutely positioned. Let me explain further:

<style>
    #outer2 {
        padding-left: 23px;
        position: relative; 
        height:auto; 
        width:200px; 
        border: 1px solid red; 
    }
    #inner2 {
        left:0;
        position:absolute; 
        height:100%; 
        width:20px; 
        border: 1px solid black; 
    }
</style>

<div id='outer2'>
    <div id='inner2'>
    </div>
</div>

As long as you don't need to support versions of Internet Explorer earlier than IE8, you can use display: table-cell to accomplish this:

HTML:

<div class="outer">
    <div class="inner">
        <p>Menu or Whatever</p>
    </div>
    <div class="inner">
        <p>Page contents...</p>
    </div>
</div>

CSS:

.inner {
    display: table-cell;
}

This will force each element with the .inner class to occupy the full height of its parent element.

I made an example resolving your problem.

You have to make a wrapper, float it, then position absolute your div and give to it 100% height.

HTML

<div class="container">
    <div class="left">"Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum." </div>
  <div class="right-wrapper">
    <div class="right">"Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua." </div>
  </div>
  <div class="clear">&nbsp;</div>
</div>

CSS:

.container {
    width: 100%;
    position:relative;
}
.left {
    width: 50%;
    background-color: rgba(0, 0, 255, 0.6);
    float: left;
}
.right-wrapper {
    width: 48%;
    float: left;
}
.right {
    height: 100%;
    position: absolute;
}

Explanation: The .right div is absolutely positioned. That means that its width and height, and top and left positiones will be calculed based on the first parent div absolutely or relative positioned ONLY if width or height properties are explicitly declared in CSS; if they aren't explicty declared, those properties will be calculed based on the parent container (.right-wrapper).

So, the 100% height of the DIV will be calculed based on .container final height, and the final position of .right position will be calculed based on the parent container.