From The Tolkien Companion, under Barrow Downs:
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There, the early Men had buried the noble among their dead. Their descendants, in the Dúnedain kingdoms of Arnor and Arthedain, therefore revered the mounds (which they knew as Tyrn Gorthad), and later princes of Arthedain and Cardolan inhumed their own royal dead there in similar months. But after the final fall of the North-kingdom, evil spirits from faraway Angmar entered the mounds and animated the long-dead corpses. These Barrow-wights walked in the hollow places among the stone-rings and the Downs became a place of dread in nearby lands.
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In case you don't know, Arthedain and Cardolan are names of two of the three kingdoms into which Arnor was split after a dispute among the sons of Arnor's king Eärendur. Since these were burial mounds for the noble and royal, it makes sense (well, kinda) to be buried with some of their riches.
The Unfinished Tales says that the wights were sent to the Downs by the Witch-King. This would imply some relation to the Wraiths, though how these people became spirits is not known, while the Wraiths obviously became such from the evil powers of Sauron.