In dogs, there are four genes on the S (spotting) locus that influence the amount of white on a given dog.  In order of most to least dominant they are:

S - solid color.  It's called "solid" but often these dogs do have small white snips on their muzzles, white spots on the chest or front of the nec, white toes,  and/or white tail tips.  This is caused  by polygenic modifiers.

s(i) - Irish spotting.  Usual markings are white collar, white blaze on the face, white socks and/or leggings, white tail tip, white chest,  white on front of neck.  Basenjis are typically Irish marked.

s(p) - particolor or piebald.  Markings vary from a blanket of color  over the body with the rest white, to varied shape/size spots on the body. Usually there are substantial spots of color on the head and rump, because the melanocytes arise from neural crest cells.  

s(w) -  extreme white spotting. White body.  Some small spots of  color are almost always seen on the head, and sometimes on the rump
as well.  Ticking is very common.  Nose and eyerims are usually pigmented as are the pads of the feet.

Greyhounds have all of these alleles, though the Irish spotting one is rare in the breed.

White Greyhounds are homozygous s(w)s(w).  This gene is seen in many other breeds, including dalmatians, bull terriers, and American bulldogs, and have the accompanying faults and problems (which may include deafness).  Occasionally one or both eyes are blue or have a blue 'chip' in the iris.

See: Clarence Little's The Inheritance of Coat Color in Dogs. p. 137 -- "As for the spotted varieties, S (solid-colored), s(i) (Irish), s(p) (piebald), and s(w) (extreme-piebald) are probably  present.  There has been no particular selection for or  against any of these varieties and, as a result, they have all been preserved. "

D.C. Warren, "Coat Color Inheritance in Greyhounds" in Journal of Heredity 1927 v.18, p513-22.  This quote is from p.513-14, under the column heading "Colors": "Five solid colors and two patterns are found in this breed of dogs.  The colors are black, blue, red, fawn, and white.  The two patterns are brindle and white spotting." [...] "Entirely white individuals are seldom found and all white Greyhounds show some pigment on the muzzle and eyes.  It is also probable that some individuals listed as white carry a small amount of color which is disregarded in registration.  As will be given later in the  discussion, it is the belief of the writer that there is no such genotype as self white in Greyhounds."

ibid. p. 517, under the column heading, "White": "An all-white individual is seldom recorded, and although crosses of white by practically all other colors were listed, very few of the  offspring are described as white.  It is also noted (Table I) that  from crosses in which one parent was white, a large number of spotted offspring were produced.  Crosses of black by white  produced only 6 white out of 85 offspring: white by brindle. 4 out of 91:  white by fawn. 6 out of 60: white by red. none out of 36: and white by blue. no white appeared among 38 young.  In all of these matings practically half of the offspring were spotted. This leads one to suspect that dogs described as white are  really spotted ones in which the colored area is reduced to a minimum."   "Self white does not seem to exist--the few white individuals apparent being cases of extreme white spotting.  Note that in this white individual the ears are pigmented--in nearly all cases white greyhounds show small pigmented areas."

ibid. p. 520 under, "Summary and Conclusions":
"5.  White spotting is recessive to self color.
6.  The extent of the white area in white spotted Greyhounds is
dependent on genetic factors, the more extended white
condition being recessive to the more restricted.
7. White greyhounds are genotypically white spotted individuals
in which the white area has been extended to a maximum."