halftone screen
A halftone screen is the grid of tiny dots a press builds a shade out of.
In more detail
Ink is either on the paper or it is not: a press has no half-black. So a grey is printed as a grid of black dots, small ones for a light grey and fat ones for a dark, close enough together that the eye reads a flat tone rather than the dots. Hold a printed photograph under a magnifying glass and the grid is there.
Three things describe such a grid. How many cells to the inch it runs to — the finer it is, the smoother the tone and the better the paper has to be. Which way it is turned, which matters because each ink has a grid of its own and two grids at the same angle land on top of one another and blotch. And the shape a dot grows into as the tone darkens: round, oval, square, a line.
What it does on a real document
The printing works usually decides all of it, and a document that says nothing lets the machine keep its own settings, which is the ordinary case. Where a job does need to say it, a declared graphics state sets `halftone`: one screen for every ink, or one per named ink with a default for the rest, each screen carrying its `frequency`, its `angle`, its `spot` shape and whether the device is asked for its precise algorithm.
None of it shows on a screen. It is read by the machine that lays the ink down, and by nothing else, which is why a document can carry it and look exactly the same on every display.
Where it comes from, and why
Photographs could not be printed at all until somebody thought of shooting them through a ruled glass grid, in the second half of the nineteenth century: the grid turned a continuous tone into dots a plate could hold. The glass went, the ruling stayed, and the vocabulary — frequency, angle, dot shape — is still the one the trade uses.