Q:

Floating elements within a div, floats outside of div. Why?

Say you have a div, give it a definite width and put elements in it, in my case an img and another div. 
The idea is that the content of the container div will cause the container div to stretch out, and be a background for the content. But when I do this, the containing div shrinks to fit the non-floating objects, and the floating objects will be either all the way out, or half out, half in, and not have any bearing on the size of the big div.
Why is this? Is there something I'm missing, and how can I get floated items to stretch out the height of a containing div?

A:

The easiest is to put overflow:hidden on the parent div and don't specify a height:
#parent { overflow: hidden }

Another way is to also float the parent div:
#parent { float: left; width: 100% }

Another way uses a clear element:
<div class="parent">
   <img class="floated_child" src="..." />
   <span class="clear"></span>
</div>

CSS
span.clear { clear: left; display: block; }

A:

Reason
The problem is that floating elements are out-of-flow:

An element is called out of flow if it is floated, absolutely
  positioned, or is the root element.

Therefore, they don't impact surrounding elements as an in-flow element would.
This is explained in 9.5 Floats:

Since a float is not in the flow, non-positioned block boxes created
  before and after the float box flow vertically as if the float did not
  exist. However, the current and subsequent line boxes created next to
  the float are shortened as necessary to make room for the margin box
  of the float.

html {
  width: 550px;
  border: 1px solid;
}
body {
  font-family: sans-serif;
  color: rgba(0,0,0,.15);
}
body:after {
  content: '';
  display: block;
  clear: both;
}
div {
  position: relative;
}
div:after {
  font-size: 200%;
  position: absolute;
  left: 0;
  right: 0;
  top: 0;
  text-align: center;
}
.block-sibling {
  border: 3px solid green;
}
.block-sibling:after {
  content: 'Block sibling';
  color: green;
}
.float {
  float: left;
  border: 3px solid red;
  height: 90px;
  width: 150px;
  z-index: 1;
}
.float:after {
  content: 'Float';
  color: red;
}
<div class="float"></div>
<div class="block-sibling">
  Lorem ipsum dolor sit amet, consectetur adipiscing elit. Donec a diam lectus. Sed sit amet ipsum mauris. Maecenas congue ligula ac quam viverra nec consectetur ante hendrerit. Donec et mollis dolor. Praesent et diam eget libero egestas mattis sit amet vitae augue. Nam tincidunt congue enim, ut porta lorem lacinia consectetur. Donec ut libero sed arcu vehicula ultricies a non tortor.
</div>

This is also specified in 10.6 Calculating heights and margins. For "normal" blocks,

Only children in the normal flow are taken into account (i.e.,
  floating boxes and absolutely positioned boxes are ignored […])

html {
  width: 550px;
  border: 1px solid;
}
body {
  font-family: sans-serif;
  color: rgba(0,0,0,.15);
}
body:after {
  content: '';
  display: block;
  clear: both;
}
div {
  position: relative;
}
div:after {
  font-size: 200%;
  position: absolute;
  left: 0;
  right: 0;
  top: 0;
  text-align: center;
}
.block-parent {
  border: 3px solid blue;
}
.block-parent:after {
  content: 'Block parent';
  color: blue;
}
.float {
  float: left;
  border: 3px solid red;
  height: 130px;
  width: 150px;
}
.float:after {
  content: 'Float';
  color: red;
}
<div class="block-parent">
  <div class="float"></div>
  Lorem ipsum dolor sit amet, consectetur adipiscing elit. Donec a diam lectus. Sed sit amet ipsum mauris. Maecenas congue ligula ac quam viverra nec consectetur ante hendrerit.
</div>

Hacky solution: clearance
A way to solve the problem is forcing some in-flow element to be placed below all floats. Then, the height of the parent will grow to wrap that element (and thus the floats too).
This can be achieved using the clear property:

This property indicates which sides of an element's box(es) may not
  be adjacent to an earlier floating box.

html {
  width: 550px;
  border: 1px solid;
}
body {
  font-family: sans-serif;
  color: rgba(0,0,0,.15);
}
body:after {
  content: '';
  display: block;
  clear: both;
}
div {
  position: relative;
}
div:after {
  font-size: 200%;
  position: absolute;
  left: 0;
  right: 0;
  top: 0;
  text-align: center;
}
.block-parent {
  border: 3px solid blue;
}
.block-parent:after {
  content: 'Block parent';
  color: blue;
}
.float {
  float: left;
  border: 3px solid red;
  height: 84px;
  width: 150px;
}
.float:after {
  content: 'Float';
  color: red;
}
.clear {
  clear: both;
  text-align: center;
  height: 37px;
  border: 3px dashed pink;
}
.clear:after {
  position: static;
  content: 'Block sibling with clearance';
  color: pink;
}
<div class="block-parent">
  <div class="float"></div>
  Lorem ipsum dolor sit amet, consectetur adipiscing elit. Donec a diam lectus. Sed sit amet ipsum mauris. Maecenas congue ligula ac quam viverra.
  <div class="clear"></div>
</div>

So a solution is adding an empty element with clear: both as the last sibling of the floats
<div style="clear: both"></div>

However, that is not semantic. So better generate a pseudo-element at the end of the parent:
.clearfix::after {
  clear: both;
  display: block;
}

There are multiple variants of this approach, e.g. using the deprecated single colon syntax :after to support old browsers, or using other block-level displays like display: table.
Solution: BFC roots
There is an exception to the problematic behavior defined at the beginning: if a block element establishes a Block Formatting Context (is a BFC root), then it will also wrap its floating contents.
According to 10.6.7 'Auto' heights for block formatting context roots,

If the element has any floating descendants whose bottom margin edge
  is below the element's bottom content edge, then the height is
  increased to include those edges.

html {
  width: 550px;
  border: 1px solid;
}
body {
  font-family: sans-serif;
  color: rgba(0,0,0,.15);
}
body:after {
  content: '';
  display: block;
  clear: both;
}
div {
  position: relative;
}
div:after {
  font-size: 200%;
  position: absolute;
  left: 0;
  right: 0;
  top: 0;
  text-align: center;
}
.block-parent {
  border: 3px solid blue;
}
.block-parent.bfc-root:after {
  content: 'BFC parent';
  color: blue;
}
.float {
  float: left;
  border: 3px solid red;
  height: 127px;
  width: 150px;
}
.float:after {
  content: 'Float';
  color: red;
}
.bfc-root {
  overflow: hidden;
}
<div class="block-parent bfc-root">
  <div class="float"></div>
  Lorem ipsum dolor sit amet, consectetur adipiscing elit. Donec a diam lectus. Sed sit amet ipsum mauris. Maecenas congue ligula ac quam viverra nec consectetur ante hendrerit.
</div>

Additionally, as explained 9.5 Floats, BFC roots are also useful because of the following:

The border box of a table, a block-level replaced element, or an
  element in the normal flow that establishes a new block formatting
  context […] must not overlap the margin box of any floats in the same
  block formatting context as the element itself.

html {
  width: 550px;
  border: 1px solid;
}
body {
  font-family: sans-serif;
  color: rgba(0,0,0,.15);
}
body:after {
  content: '';
  display: block;
  clear: both;
}
div {
  position: relative;
}
div:after {
  font-size: 200%;
  position: absolute;
  left: 0;
  right: 0;
  top: 0;
  text-align: center;
}
.block-sibling {
  border: 3px solid green;
}
.block-sibling.bfc-root:after {
  content: 'BFC sibling';
  color: green;
}
.float {
  float: left;
  border: 3px solid red;
  height: 90px;
  width: 150px;
  z-index: 1;
}
.float:after {
  content: 'Float';
  color: red;
}
.bfc-root {
  overflow: hidden;
}
<div class="float"></div>
<div class="block-sibling bfc-root">
  Lorem ipsum dolor sit amet, consectetur adipiscing elit. Donec a diam lectus. Sed sit amet ipsum mauris. Maecenas congue ligula ac quam viverra nec consectetur ante hendrerit. Donec et mollis dolor. Praesent et diam eget libero egestas mattis sit amet vitae augue. Nam tincidunt congue enim, ut porta lorem lacinia consectetur.
</div>

A block formatting context is established by

Block boxes with overflow other than visible, e.g. hidden
.bfc-root {
  overflow: hidden;
  /* display: block; */
}

Block containers that are not block boxes: when display is set to inline-block, table-cell or table-caption.
.bfc-root {
  display: inline-block;
}

Floating elements: when float is set to left or right.
.bfc-root {
  float: left;
}

Absolutely positioned elements: when position is set to absolute or fixed.
.bfc-root {
  position: absolute;
}

Note those may have undesired collateral effects, like clipping overflowing content, calculating auto widths with the shrink-to-fit algorithm, or becoming out-of-flow. So the problem is that it's not possible to have an in-flow block-level element with visible overflow that establishes a BFC.
Display L3 addresses these issues:

Created the flow and flow-root inner display
  types to better express flow layout display types and to
  create an explicit switch for making an element a BFC root.
  (This should eliminate the need for hacks like ::after { clear: both; } and overflow: hidden […])

Sadly, there is no browser support yet. Eventually we may be able to use
.bfc-root {
  display: flow-root;
}

A:

Put your floating div(s) in a div and in CSS give it overflow:hidden;
it will work fine.

