Halftone screening, also sometimes called halftoning, screening or dithering, is effectively a technique to reliably produce optical illusions that fool the eye into seeing tones and colours that are not actually present on the printed matter.   

Most printing technologies are not capable of printing a significant number of different levels for any single colour. Offset and flexo presses, for instance can only place ink or no ink. Halftone screening is a method to make it look as if many more levels of gray are visible in the print, by laying down ink in some areas and not in others, but using such a small pattern of dots that the individual dots cannot be seen at normal viewing distance.
The conventional halftone screening process breaks a continuous tone black and white image into a series of dots of varying sizes and places these dots in a rigid grid pattern. Smaller dots give lighter tones and the dot sizes within the grid are increased in size to give progressively darker shades until the dots grow so large that they overlap with adjacent dots to form a solid of maximum density (100%).
This concept, of varying the dot sizes on a fixed-pitch grid, is generally called “conventional screening,” or more technically, amplitude modulation (AM). The Amplitude Modulation terminology (and Frequency Modulation, described below) are familiar terms from the world of radio. At longer wavelengths (long wave and medium wave), the strength of the signal (the amplitude) varies while the frequency remains constant. At shorter wavelengths (FM radio) the strength of the signal remains constant, but the frequency is varied in other words, is modulated.
The amount of detail that can be shown is controlled by changing the screen ruling, or the constant distance between the dot centers, measured in lines per inch. The closer the dots in a particular screen (the higher the screen ruling), the better the representation of image detail, but the more difficult it may be to print or reproduce.
For a colour image it is common to break the colours down into the process colour inks – usually cyan, magenta, yellow, and black (a process known as separation). Each of these is screened separately using techniques designed to minimize any interaction between the colourants.

Halftone screening for digital print
Although these AM screens made the transition to high resolution laser film and plate imagers very well, they are less applicable to digital printing systems, as those have entirely different imaging characteristics. To generalize, digital presses tend to print at lower resolutions but with a greater ability to reproduce very small (“highlight”) imaging spots. Screens therefore needed to be developed for digital printing to take best advantage of these limitations and benefits. In production printing, these digital screens are mostly the Frequency Modulation (FM) type, also called stochastic screens.
In very basic terms, an FM screen uses dots that are all the same size, but the distance between them is varied to give lighter or darker tones. There is no regular grid placement (unlike conventional AM screens). In fact the placement is more or less randomized (which is what the word ‘stochastic’ means), but truly random placement leads to a very ‘noisy’ result with uneven tonality, so the placement algorithms are carefully set to avoid this. 
FM screening has the benefit that it avoids moiré patterning when using multiple colours of ink on top of each other, which can be a problem with AM screens. An AM screen can retain more detail and tonal subtlety when rendered at higher line screens and higher resolutions … at least until the physical capabilities of reproduction on the printing press hide those gains, or even lose the smaller halftone dots completely. 
On the other hand, FM screening can retain more detail and tonal subtlety than AM screening can at the same resolution. This makes FM screens particularly relevant to single-pass inkjet presses, which tend to have lower resolutions than the imaging methods used on, say, offset lithography. An AM screen at 600 dots per inch (dpi) would be very visible from a reading distance of less than a metre or so, while an FM screen can use dots that are sufficiently small that they produce the optical illusion that there are no dots at all, just smooth tones. 
 Global Graphics has developed an advanced FM screening technology that is already widely used in offset lithography and screen printing. The Global Graphics Harlequin Dispersed Screening™ (HDS) a second-generation FM screening algorithm that provides a high quality alternative to conventional screening in many situations. 

Modern single-pass inkjet presses
Inkjets work, as the name suggests, by projecting ink. Current print heads generate drops of ink with very fine control over their size/volume.
Originally all inkjet print heads were binary, meaning they only generate one, consistent size of drop. Binary in this case means all or nothing – there’s a full-sized drop on the media, or nothing at all. 
More recently, many print head developers have created multi-level print heads that can place any one of several different amounts of ink at a single location on the media. This can be done by generating single drops of different sizes, or by jetting multiple drops onto the same location.
Different heads and different configurations can deliver a variety of different amounts of ink; from two levels upwards, although the maximum common number is about five. The main effect is that for the same nozzle pitch, native resolution and speed, it is possible to print drops that give different image densities on the substrate. Assuming a coloured ink, small drops generate “light” tones and larger drops generate “darker” tones.
A common industry description for multi-level print heads is “grayscale heads,” which describes the effect of printing black ink with different sized drops: the smaller drops appear gray, and larger drops give progressively darker grays until black is reached. However, note that grayscale heads can be used to print any colour, or indeed clear, white and other ink types.

Selecting drop sizes
With multi-level screens, the smallest ink drop sizes are used for the lightest tones, and the largest drops for the dark tones through to solids. Simply dividing the tonal range from 0 to 100% ink coverage by the number of drop sizes the heads can deliver doesn’t yield good results, however. 
In practice 100% coverage of the lightest drops usually produces a tone that’s darker than 50% of the tonal range, sometimes up to 80%. If three or four drop sizes are used, then the three larger sizes will normally all be used for the tones that are darker than 50 to 60%.
The tonal range for which each drop size is used is also overlapped fairly broadly, with some coverage of the larger sized drops being introduced before the maximum coverage of the smallest drop size is reached. 
If the tonal ranges are not overlapped there can be a visible change in texture at the point of transition from one drop size to the next.