Barrow-downs or Tyrn Gorthad were a series of low hills east of The Shire, behind the Old Forest, and west of the village of Bree. Many of the hills were crowned with megaliths and barrows, whence their name.
The Barrow-downs were first inhabited by Men related to the Edain in the First Age, together with the Hills of Evendim to the north. They fled east as Easterlings invaded Eriador and passed on to Beleriand, but after these had left or been killed in the War of Wrath the Edain returned to their old homes.
During the Second Age they were fairly numerous, and when they met with the Númenóreans the Barrow-downs were the first places where the Dúnedain emigrees from Númenor settled. When Elendil returned to Middle-earth, the Barrow-downs were incorporated in the kingdom of Arnor.
After the split of Arnor the Barrow-downs became the capital of Cardolan. After Rhudaur fell to Angmar, the Dúnedain of Cardolan entrenched themselves here, but eventually the realm fell. The Barrow-wights were now sent there by the Witch-king. After Arthedain once again claimed the kingship over all of Arnor the Dúnedain tried to recolonize Cardolan, but this failed because of the Barrow-wights.
A possible real-life inspiration for Tolkien were the Barrow Downs of Warwickshire, near the village of Long Compton.
What follows is from Tolkien's the book The Fellowship of the Ring.
During the War of the Ring Frodo Baggins and company were trapped in the downs by a wight, probably in the same cairn which held the grave of the last prince of Cardolan. They were rescued by Tom Bombadil, and there got their swords.