The method of putting a copyright, or any other data for that matter, into an image file is a process or science called steganography. If you are interested in more info on the theory behind it here is a link:
http://www.jjtc.com/Steganography/
For those who don't want to read a volume of material I will give you the shorter answer. There are many details I'm going to leave out so those techie's out there please don't go slamming me for ingnoring something. What I will cover are the basic's of how this is accomplished.
A digital picture that has been stored to a file is simply a stream of numbers that describe the color and intensity of each pixel that is in the picture. I will use for an example, a black and white picture that is a resolution 640x480 that has been taken with 8 bit resolution. That picture has 640x480=307200 pixels that describe the picture. The numbers mean 640 pixels or dots per row with 480 rows. The intensity of each pixel is represented by an 8 bit number, which for our example is a value between 0 and 255. For this case lets let 0 mean white and 255 means black. Every other value from 1 to 254 is a "blend" between white and black. When you view a digital picture on a web page the software in your computer reads the picture file, it knows how to sort out the numbers in the proper order and displays your picture.
So how do I add a watermark? Well, the human eye is sensitive but not so much that you can easily determine the difference between an intensity value of 0 and 1. You will definitely see the difference between 0 and 255 but maybe not 0 and 1. Using specially designed software, you can go into a picture file and, in a predetermined manner, go in and change all of the very small numbers by one value. That is to say the software would change a 0 to a 1, or a 1 to a 2. This is done in such a fashion as to encode your message. Let's say for example I want to put the message "Copyright 2003, Phil Hoppes" into my picture. The software first breaks down the text into its numbers. (Computers don't work with letters and characters. All of these are represented by numbers.) The software then uses an algorithm to encode these text characters across all of the minor changes it makes to the picture. For example, I don't remember ASCII code but lets assume C = 003. The software would encode the number 003 across the changes in the first series of bits in the picture that it decided to change.
The point is, text can be added to a picture (or an audio file, or movie file) in such a manner that to the naked eye, the picture has not changed.
Here are a few caviats however. If the physical size of the picture is small compared to the size of the text that is desiered to add, you will completely destroy the picture in the process. For example, if you have a 25K jpg file and you want to add a 50K text message, the software may do it but you won't even recognize the picture. Since a watermark is a very small message (my message above would be only 216 bytes) and picture is say 35K or 35,000 bytes, my message is very small compared to the size of the picture. You could add my message to a picture and not corrupt the image.
Least you all think that this is an absolutly secure way to "mark" your pictures, it is not. There is software that can detect that a watermark has been added to a photo. Depending on the method, it can even remove it and restore the picture back to its original state. Like anything else, this just boils down to time and money.
For 99% of us adding a watermark is probably a good thing in that most casual and probably even a fair number of concentrated crooks may not be aware that you have added a watermark to your picture. You can get software for free that will do it. You can pay (Digimark) to have this done. All versions of Photoshop since I think 5.0 and on will do this and link you to Digimark so you can pay to have this done. You have to assess the risk vs the reward.
Hope this answers a few questions and explains the process.
Phil
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