This term describes the oathbreakers to the Kingdom of Gondor who come at last, as ghosts, to help Aragorn fight back the allies of Sauron. Back in the day, the King of the Mountains swore an oath to Gondor to help in the war against Sauron. When Isildur called in this oath, they broke their word (in fact, they had come to worship Sauron in these later days). Isildur cursed them to join the War of the Ring right at the end. Then they would be summoned by his heir to fight as dead men.
Hearing this curse, the men of the mountains grow so frightened that they try to run away from Isildur: "they hid themselves in secret places in the mountains and had no dealings with other men, but slowly dwindled in the barren hills" (5.2.90). But the curse still holds, and their ghosts aren't allowed to rest until Aragorn comes to the Stone of Erech to demand that they help him out in battle.
The Dead march with Aragorn, the Rangers, Legolas, and Gimli all the way to Pelargir, at the Mouths of Anduin. They fight the main fleet of Southrons, who have arrived in ships from Umbar and Harad (countries to the far south of Gondor). The Dead do not need physical weapons, because their main tool is fear: the Southrons are so terrified by these ghost soldiers that they throw themselves into the sea or run away. And once the Southrons have been defeated, Aragorn frees the Sleepless Dead to move on to the next life.
As Legolas points out, it's ironic that Aragorn overcomes the troops of Mordor using the main Mordor weapon of "wraiths of fear and darkness" (5.9.50). But this strategy fits with the general message of The Lord of the Rings, which is that evil is essentially self-destructive. Sauron uses fear as a weapon? Well then, his forces will be destroyed by fear. What goes around comes around, buddy.
Another example of this logic is the darkness Sauron spreads over the land when he's gearing up to attack Gondor. It's a sign of his strength, sure, but it also gives cover to the Riders of Rohan to conduct a surprise attack on the Southrons attacking Minas Tirith.