As you can see from the Wikipedia article's title, the land bridge is known by different names in different religious traditions. The name Adam's Bridge dates back to an Islamic legend that tells how the first man, Adam, walked to a mountain in Sri Lanka, called Adam's Peak, where he stood on one foot on the mountain for a thousand years as penance; more about that legend here: Adam's Peak.
[Notes by LKG]
This story is part of the Ramayana unit. Story source: Indian Myth and Legend by Donald A. Mackenzie (1913).
The War with Ravana
Rama performed sacrifices to propitiate the God of Ocean so that the Vanar forces might he enabled to pass over to Lanka, but these proved to be unavailing. Then angrily he seized his bow and shot celestial weapons into the bosom of the deep. The earth and the sea were immediately convulsed, and darkness covered the heavens; lightning flashed and thunder bellowed aloud; the mountains began to break in pieces. Rama next seized a fiery dart and threatened to dry up the waters of the sea.
At that moment the King of Ocean rose serenely above the weltering billows in all his splendour, attended by shining water snakes. He addressed Rama with great reverence, reminding him that according to ancient laws he must remain unfordable, but counselling him the while to seek the aid of the Vanar chief Nala, son of Vishwakarma, the divine artisan, so that a bridge might be constructed to enable the armies to cross the deep. Then the King of Ocean vanished amidst the waves and the heavens brightened again.
Image source: Building the bridge to Lanka
Nala and the Bridge
Nala was immediately called upon to give his aid. Assisted by his workmen, this wonderful Vanar, whose body was green, constructed a causeway of rocky islands between the mainland and Lanka (Ceylon), and to this day it is called "Rama's Bridge."
Rama meanwhile set up the Linga symbol of the god Shiva, and worshipped it on that holy island which hath since been called Ramisseram.
In five days the strait was spanned. Then Rama mounted on the back of Hanuman, son of the wind god Vayu, and Lakshmana mounted the back of Angada, son of Bali and grandson of Indra, and led the Vanar hosts across the sea.
The apes and bears which composed the great attacking army leapt from island to island, shouting: "Victory to Rama!" "Victory to Lakshmana!" "Victory to Sugriva!" Now the apes were of many colours; they were white and black, green and blue, yellow and red and brown. Sugriva shone like silver, Angada resembled a white lotus; Nila, son of Agni, was red, and Hanuman was yellow as pure gold; Sarambha had also a yellow body, and Nala was green, while Darvindha had a black body, a red face, and a yellow tail. These were all leaders and great warriors of the Vanar host.
The army landed in Lanka unopposed, and encamped on a plain fronting the capital of the Rakshasa king.
The Armies Meet
The Rakshasas issued forth speedily to attack the apes, and the blowing of horns and beating of drums sounded like to the mighty thunder peals at a Yuga end. Indrajit was the Rakshasa leader. His followers rode on elephants and lions, on camels and asses, on hogs and hyenas, and on wolves; they were armed with bows and arrows, maces, spears, tridents, swords, and beams, but some had also magic weapons. Roaring and swaying, they drove forward like to long sea-rollers assaulting the shore.
The gigantic apes wielded trees for clubs and threw great boulders, but some depended on their sword-like nails and their long arrowy teeth. They rushed against the demons, shouting "Rama, Rama!" and soon the plain was covered by heaps of writhing bodies and severed limbs, while rivers of blood streamed across it from between the battling hosts. Rama looked on without fear. He reposed his faith on the apes, for he knew that they were incarnations of the gods.
Indrajit, Son of Ravana, and Kumbakarna, Ravana's Brother
The apes were driven back until Sugriva flung a great tree, which shattered the chariot of Indrajit. Then the Rakshasa leader and his army took flight.
Indrajit obtained a new chariot by offering up in sacrifice a black goat, and, returning to the battlefield with his forces, he shot arrows at Rama and Lakshmana. Then he threw a serpent noose, which bound the two brothers so that they were unable to move. Great was their peril, but Vayu, god of wind, sent to their aid the great celestial bird Garuda, the serpent killer, and the snakes which formed the noose fled from before it, whereat the brethren, who had meantime fallen in a swoon, rose up again.
Ravana then came forth, but Rama shot arrows which swept the ten crowns from his ten heads, and he retired in his shame and skulked in the city.
The Rakshasas were in desperate straits and bethought them to awaken Kumbhakarna, the mightiest of all the demons. In former days, he had terrorized the Universe; he continually devoured human beings and had defeated Indra even, but Brahma intervened and decreed that he would sleep for six months and then awaken for one day only. Each time he awoke he devoured a great meal, after which he was again overpowered by slumber.
Thousands of men danced and shouted and blew trumpets beside the great sleeper, but he could not be wakened; elephants were driven over his body, yet he never moved; then beautiful women came and caressed him, and he suddenly opened his eyes and roared like to the sea. His eyes were red with anger, and he cried: "Why have I been awakened before my time?"
The Rakshasas informed Kumbhakarna of the army which surrounded the city, and they brought him much food; greedily he swallowed swine and deer and many human beings and drank rivers of wine. Refreshed, but not yet satisfied, he arose and said: "Where are the apes so that I may devour them?"
Next: Rama Confronts Ravana