Double-knitting is knitting both sides of an item at the same time. If you are familiar with intermediate-to-advanced knitting, you might recognise it from knitting two socks at the same time. I am referring to the colour-work kind, where you are knitting a reversible image.
KnittingHelp.com has a good pattern, the Heart Hot Pad, that shows what I'm thinking of here.
The downside is, it is mirror-image reversible. Take this image and fold it in half down the center — that would be the result of doing normal colour-work double-knitting. This is fine for many things, but not when I want to knit, say, letters, or logos.
This page assumes you are familiar with the following knitting concepts:
Each link in that list goes to a page with a video. The text links go to sites with text and accompanying images.
(Incidentally, if you'd like assistance on reading a knitting chart, this document is wicked useful.)
You work a chart by reading it from bottom to top, right-to-left on odd rows, and left-to-right on even rows.
I'll go into how to double-knit at all first, just to be thorough.
I'll be referring to this chart:

A 22r x 17s blue and white knitting chart of a spade.
When you cast on, cast on with both strands at the same time.
Also, cast on twice as many stitches as the pattern calls for. For example, in the chart above, the pattern is 17 stitches wide, so you would cast on 34 stitches.
The first row is always, in my experience, kind of a pain, because you need to pick out the colour to knit into, and sometimes the strands are twisted, so this takes a little finagling. It may be better to take the time to cast on with the colours alternating correctly.
I typically have a border in my patterns, so that I have at least one row/column (times two, I mean) where the stitches are normal.
You are knitting an item facing you, and purling its opposite facing away from you, at the same time. So for each row, knit one in the colour facing you, and purl one in the colour facing away from you. This means you will be doing "k1/p1 to end" for each row.
When you knit and purl, you will carry both strands back and fore (respectively), but you will only knit or purl with the intended colour.
Proper edging is required, because otherwise, the work will gape open on the sides. To do the edging:
The chart itself is showing one of the two items that you are knitting. In this case, it is showing the one where the spade is white and the background is blue.
Each cell in a chart represents two stitches: the knitted main colour, and the purled other colour. When I knit row 1 of the chart above, I will be doing the following:
The problem comes when you get to the even row, and the colours are reversed. Because you are no longer knitting the white spade on blue background item — now, you are knitting the blue spade on white background. Therefore, for row 2, I would do the following:
As stated below, you are knitting the item that is facing you, and purling the item that is away from you, so the colours flip.
I will likely design something that will allow you to click a button for "I'm on an even/odd row" and it will change the colours accordingly.
To cast off, work a row with k1/p1 as though you were just doing another row, but cast off as you go.
For this section, let's switch charts. Let's say you wanted to knit the number 3, using double-knitting:

A 21r x 14s blue and white letter 3.
You would want the reverse side to look like this when you look at it:

A 21r x 14s white and blue letter 3.
However, double-knitting a non-symmetrical item (like, say, the number 3) would instead result in seeing this:

A 21r x 14s white and blue letter 3, reversed horizontally.
I mean, unless you wanted to knit a curly letter E.
This is due to normal double-knitting's mirror-reversibility. Put your hands flat together, and look at the back of one hand. Then, flip your hands over. That's effectively mirror-reversible double-knitting.
Now, put the back of one hand into the palm of your other, and hold your hands flat. That's what you are trying to knit. If you were physically capable of flipping your hands over while they were together, you'd see that the shape is exactly the same on both sides.
I can't go further with this without a video, or perhaps semaphore flags.
To work through this, continue to follow this instruction directly: Knit the item facing you, and purl its opposite facing away from you.
Let's say you are four rows in, meaning the main colour is white and the secondary colour is blue*. You've worked the first four stitches thusly:
* If you have an easier way of indicating primary/secondary colours, obviously go with what works for you. This is just what works for me.
But then you get to the fifth stitch. In the white-on-blue chart, the fifth stitch of the fourth row is blue, so you would assume you'd be knitting a blue stitch and purling a white stitch. However, in the reversed blue-on-white chart — in the item that is facing away — the stitch is also blue.
So, for that cell, you knit one blue, and purl one blue.
What if the first chart was blue-on-white, and the last was white-on-blue?


Then row 4, stitch 5 would be knit 1 white, and purl 1 white.
Interestingly, I make a knitting chart that covers for this eventuality.
A sample knitting chart for this:

21r x 14s, truly-reversible double-knitting chart, with legend.
You can see what I mean, for row 5, stitch 4.
With regards to the "What If…?" posed above, if I changed col1 and col2 to their opposite colours, it would still work.
I actually have an Excel workbook that does most of this for me, because trying to think it through makes my eyes cross. You can download it here.
WARNING: The file is an Excel (97-2003 compatible) workbook with macros. I make no promises about it at all. Also, you need to be able to run macros in order to use that workbook. Microsoft has a pretty useful page about macro security, and how to change the settings.