'Halftone screen' is the way printers have traditionally put photographs on paper, particularly in the time period DP is most concerned about. It works by having a number of dots in a regular pattern (usually hexagonal or square), and each dot is smaller or larger depending on how dark the overall colour of the area is. When viewed from a distance this produces the illusion of graytones, while the actual pattern on the page is just black or white without in between shades. Newspapers are still printed this way, as you can see by holding up a newspaper photo close to your eye.
This halftone process creates a problem when scanning these images: the halftone dot pattern interferes with the pixels of the computer representation to create light and dark bands of colour, espcially when the image is resized. Only people doing content-providing need to worry about this.