In addition to what Webster 1913 says below about "Vandyke" referring to a painting by Flemish painter Anthony Van Dyck (1599-1641), a shade of brown popularized in his paintings, or a collar or decorative edge resembling those seen in his paintings, a Vandyke or Van Dyke can also be a type of beard: short, neatly trimmed, and pointed/V-shaped. This type of beard was apparently fashionable enough at the time to appear frequently in Van Dyck's portraits of the aristocracy of England (he was court painter briefly for James I and longer for Charles I). Presumably the spelling difference is just a result of the non-standardized spelling in the English of the era; although Van Dyck is how the surname was spelled originally, the painter was knighted as Sir Anthony Vandyke in 1633.
A Vandyke is similar to a goatee; exactly what the difference is varies according to who you ask. The most common statement is that a Vandyke includes a mustache and a goatee does not. As one blogger put it, "The Count has a goatee . . . Colonel Sanders has a Vandyke." One author, Ken Rickard, includes a requirement for a soul patch below the lower lip, what he calls an imperial, as well as the mustache and chin hair, but he seems to be alone in this definition. (Judging by nationalbeardregistry.org, the practical difference between Vandyke and goatee is more that Vandykes are generally kept much neater and shorter than goatees.)
Also, the name Vandyke is applied in photography to a water-developed kallitype print which scomes out in a similar deep brown color to that known as Vandyke brown. These are also known as brown prints. There is also a Vandyke process in photography, a photographic development process which uses particular chemicals; Peter Marshall of photography.about.com says that the two are completely different and the process is named after F. Vandyke, a colonial civil servant in India, who invented the process to print maps.
Vandyke Software is a company specializing in computer security programs.