Un-confusing the Circle of Confusion
There are a lot of technical terms in the language of photography. Sometimes it's easy for photographers to use terms casually without really knowing what they actually mean. No photography term seems to cause more confusion as does the infamous "circle of confusion."
Take a photograph of a point source of light. If the image is in focus, you should end up with a sharp, distinct point representing that light. Throw the lens way out of focus and take another shot. That sharp point of light will now look more like a disc with a discernible diameter than a point. Depending on the optics and the circumstances it may vary in size and apparent edge blur, but it certainly won't look like a point.
But what about in between these two extremes?
To consider this in detail, take a series of shots while slowly move the lens ring ever closer back into focus. When you're finished, line the resulting images up in a row and examine them. You know that at one end of the row the images will clearly show sharp points of light and out of focus discs at the other. Somewhere along the series, the shape of the rendered light must therefore shift. Look at some of them and you'll see a point, look at others and that same subject will look like a disc. As you move toward the out-of-focus end of the series, the spot in each successive image will actually be slightly larger than the one before it, but the ones close to the in-focus end will still appear acceptably sharp for you to consider them in focus. At some point though, it will become evident that an image is no longer sharp and you will see a disc and not a point.
The point at which you really can't tell whether what you are looking at is a point or a disc is the circle of confusion. Literally, you are confused as to which you are looking at. Far from being a cryptic term, "circle of confusion" is actually quite descriptive of the phenomenon it names.
Exactly how big this is depends on how critically and how close you examine that series of images. If you stand far enough away, most if not all the images may look sharp. Look close enough or use a magnifying glass you will certainly come to a different conclusion.
The generally accepted circle of confusion value for a full-frame (35mm) sensor is 0.03 mm. That's 0.03 as measured on the sensor plane itself. If your camera's sensor is smaller, the circle of confusion value should generally be divided by the crop factor of your sensor. The exact value you use though depends on more than just generally accepted values. If you are doing more critical work, a more stringent value would undoubtedly be called for. If your work will be printed on a billboard to be viewed from street level, you can probably get away with less exacting criteria.