"A Links is any rough grassy area between the sea and the land" says this site devoted to the history of golf in Scotland. Links, when referring to a golf course, is often used as a singular noun these days, although not always - this Wikipedia page has the sentence "The Carnoustie Golf Links are in Carnoustie, Angus, Scotland".
The word link goes back to Old English. The OED defines its original sense as "rising ground; a ridge or bank". It is now obsolete in this sense, or a dialect term. In Scotland from the 15th century the plural, links, was used to mean "comparatively level or gently undulating sandy ground near the sea-shore, covered with turf, coarse grass, etc" (OED). The first OED citation for links meaning "the ground on which golf is played" is 1728 (golf is thought to have been first played in Scotland in the 15th century).
The Old English word for link was hlinc, which the OED suggests may be related to the modern verb lean. The only other word in the OED which developed from hlinc is another rare or dialect word for "a rising ground; a ridge; a ledge", namely linch. The word linch-pin (also spelled lynch-pin) comes from another Old English word, lynis. The modern German word for linch-pin is Lünse.
The other, chain-related, sense of link has a different history, and it entered English in the late 15th century. It's from Old Norse or another old Scandinavian language. There was a cognate in Old English, hlęncan, which was a plural form (of lank) and meant 'armour'.