Desktop publishing (DTP) is the process of creating printed documents that look professionally produced, using page layout software running on a personal computer, along with a high-quality, yet affordable, printer. To publish something with the traditional method, you would send typed or handwritten text to a typesetter, who would turn it into typeset text called "galleys," which took a couple of days. If there were corrections, it took another couple of days to get those back. If you didn't know how to layout the pages yourself, you'd take the galleys to a print shop, along with your art (illustrations and photographs).
The people there would cut up the galleys with scissors and paste the pieces onto the pages along with the artwork. If something needed to be changed on the finished "paste-up" or "mechanical," it would be possible, but a lengthy and expensive process. Finally, the print shop would reproduce the document in quantity. Instead, you might pay a graphic designer to take the project from conception to completion, and the designer would go through this process, creating the mechanical herself and taking it to the print shop to be reproduced. With desktop publishing, by contrast, you can create the entire document sitting at your own desk.
You can think of the page layout software and the computer as the typesetting and layout area, and the laser printer as the printing press. You proof the project on your own printer; if it isn't right, you just turn back to your computer, make the changes, and print it again. Depending on how many final copies you need and the quality you want, you can print the job yourself, duplicate the masters you print on a photocopy machine, or have a professional press reproduce them. For the ultimate in quality, you can take your disk with your finished document to a service bureau, have it output on a high-resolution imagesetter, and then have the resulting pages professionally reproduced.