Imagine a perfect white point in an empty black room. The point has no height, and no width. If you focus an optically perfect lens on that point, it forms a perfect point on the film as well. If, however, you focus slightly in front of or behind the point, the point will image on the film as a small blurry circle. If that circle is small enough, it will still look like a point when enlarged for printing. The "circle of confusion" is typically calculated as the largest on-film circle that you see as a point when you make an 8 × 12 print and view it from a "normal" viewing distance, typically 2-3 feet. Anything larger is seen as a small circle, and is therefore perceived as out of focus.
f/Calc calculates the CoC using the "Zeiss formula": d/1730, where d is the diagonal measure of the film, in millimeters. This formula yields acceptable values for most uses.
Many lenses have depth of field markings engraved on the barrel: the CoC affects DoF, so obviously these manufacturers are comfortable giving you a very fixed definition for CoC. This is because the main factor affecting CoC is the film size; lenses are designed for a particular camera, and thus a particular film size.
The film size is important because you don't have to enlarge large negatives as much to get a particular sized print. So, a 6 × 4.5 cm frame, being roughly twice the size of a 35mm frame, will have a CoC that's roughly twice the size of that for a 35mm frame. In other words, if a fuzzy disc 0.025 mm wide looks like a point when printed from 35mm film, you can have an 0.043 mm disc on 6 × 4.5 cm film and still have the same apparent degree of sharpness if you enlarge it to the same size print as you made from the 35mm frame.
Technically, the print size and the viewing distance are factors in choosing an appropriate CoC value. For most purposes, these factors are not important, because larger prints are generally viewed from farther away than smaller prints, and because people making large prints generally use large film sizes. Nevertheless, there are exceptional conditions, like prints hung in an art gallery: a gallery setting often demands large photos which will be viewed relatively close-up.
The CoC value also takes into account imperfections in the camera and enlarging lenses used, and the resolution of the film and paper used to make the image. Since most people take pictures with typical cameras and have the photos developed at generic photo labs which print their photos on standard paper with typical photo printing machines operated by technicians of average talent, it is usually not important to take these various quality issues into account: they're effectively built into the Zeiss formula. Your situation may be atypical, however. If you're using a digital camera and an inkjet printer, for example, you might need a smaller CoC to account for each device's relatively low resolution. Cheap lenses will also require smaller CoC values than sharp ones to achieve the same degree of apparent sharpness. (Smaller CoCs can only go so far in improving sharpness, however.)