Css – Header div stays at top, vertical scrolling div below with scrollbar only attached to that div

2015 update: the flexbox approach

There are two other answers briefly mentioning flexbox; however, that was more than two years ago, and they don't provide any examples. The specification for flexbox has definitely settled now.

Note: Though CSS Flexible Boxes Layout specification is at the Candidate Recommendation stage, not all browsers have implemented it. WebKit implementation must be prefixed with -webkit-; Internet Explorer implements an old version of the spec, prefixed with -ms-; Opera 12.10 implements the latest version of the spec, unprefixed. See the compatibility table on each property for an up-to-date compatibility status.

(taken from https://developer.mozilla.org/en-US/docs/Web/Guide/CSS/Flexible_boxes)

All major browsers and IE11+ support Flexbox. For IE 10 or older, you can use the FlexieJS shim.

To check current support you can also see here: http://caniuse.com/#feat=flexbox

Working example

With flexbox you can easily switch between any of your rows or columns either having fixed dimensions, content-sized dimensions or remaining-space dimensions. In my example I have set the header to snap to its content (as per the OPs question), I've added a footer to show how to add a fixed-height region and then set the content area to fill up the remaining space.

html,
body {
  height: 100%;
  margin: 0;
}

.box {
  display: flex;
  flex-flow: column;
  height: 100%;
}

.box .row {
  border: 1px dotted grey;
}

.box .row.header {
  flex: 0 1 auto;
  /* The above is shorthand for:
  flex-grow: 0,
  flex-shrink: 1,
  flex-basis: auto
  */
}

.box .row.content {
  flex: 1 1 auto;
}

.box .row.footer {
  flex: 0 1 40px;
}

<!-- Obviously, you could use HTML5 tags like `header`, `footer` and `section` -->

<div class="box">
  <div class="row header">
    <p><b>header</b>
      <br />
      <br />(sized to content)</p>
  </div>
  <div class="row content">
    <p>
      <b>content</b>
      (fills remaining space)
    </p>
  </div>
  <div class="row footer">
    <p><b>footer</b> (fixed height)</p>
  </div>
</div>

In the CSS above, the flex property shorthands the flex-grow, flex-shrink, and flex-basis properties to establish the flexibility of the flex items. Mozilla has a good introduction to the flexible boxes model.

The problem that I found with Boushley's answer is that if the right column is longer than the left it will just wrap around the left and resume filling the whole space. This is not the behavior I was looking for. After searching through lots of 'solutions' I found a tutorial (now link is dead) on creating three column pages.

The author offer's three different ways, one fixed width, one with three variable columns and one with fixed outer columns and a variable width middle. Much more elegant and effective than other examples I found. Significantly improved my understanding of CSS layout.

Basically, in the simple case above, float the first column left and give it a fixed width. Then give the column on the right a left-margin that is a little wider than the first column. That's it. Done. Ala Boushley's code:

Here's a demo in Stack Snippets & jsFiddle

#left {
  float: left;
  width: 180px;
}

#right {
  margin-left: 180px;
}

/* just to highlight divs for example*/
#left { background-color: pink; }
#right { background-color: lightgreen;}

<div id="left">  left  </div>
<div id="right"> right </div>

With Boushley's example the left column holds the other column to the right. As soon as the left column ends the right begins filling the whole space again. Here the right column simply aligns further into the page and the left column occupies it's big fat margin. No flow interactions needed.