What is the story about Shri Krishna lifting Govardhan Giri to save the villagers from heavy rain?
The answer to this can be found in Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 10.24: Worshiping Govardhana Hill and Śrīmad-Bhāgavatam (Bhāgavata Purāṇa)10.25: Lord Kṛṣṇa Lifts Govardhana Hill. Here is the story in short:
Lord Śrī Kṛṣṇa crushes the pride of Indra by prohibiting a sacrifice intended for him and initiating a substitute sacrifice in worship of Govardhana Hill.
Indra, angered at the disruption of the sacrifice meant for him and falsely presuming himself the supreme controller, said, “People often give up the pursuit of transcendental knowledge — the means for self-realization — and imagine that they can cross over the ocean of material existence by mundane fruitive sacrifices. Similarly, these cowherd men have become intoxicated by pride and have offended me by taking shelter of an ignorant, ordinary child — Kṛṣṇa.”
To remove this supposed pride of the residents of Vraja, Indra sent the clouds known as Sāṁvartaka, whose function is to facilitate the destruction of the world. He sent them to harass the Vrajavāsīs with downpours of rain and hail. The cowherd community was very disturbed by this and approached Kṛṣṇa for shelter.
Understanding that this trouble was the work of Indra, Kṛṣṇa decided to smash to pieces Indra’s false prestige, and thus He lifted Govardhana Hill with one hand. He then invited the entire cowherd community to take shelter in the dry space beneath the mountain. For seven successive days He held up the hill, until Indra finally understood Kṛṣṇa’s mystic power and ordered the clouds to withdraw.
Once the people of Vrindavana were conducting a Yagna, and Krishna persuaded them to stop, suggesting that they worshiped the Govardhana mountain instead. When Indra found out about this, he was furious, and he sent to Samvartaka clouds, which usually only appear in the Pralaya to destroy the world, to destroy the people of Vrindavana. The people turned to Krishna for help, and he lifted the Govardhana mountain with one hand to give people shelter from the rain. He held it for seven days until Indra realized the power of Krishna and sent the clouds away, as described in this chapter of the Srimad Bhagavatam.
On Indra’s order the clouds of universal destruction, released untimely from their bonds, went to the cowherd pastures of Nanda Mahārāja. There they began to torment the inhabitants by powerfully pouring down torrents of rain upon them. Propelled by the fearsome wind-gods, the clouds blazed with lightning bolts and roared with thunder as they hurled down hailstones. As the clouds released torrents of rain as thick as massive columns, the earth was submerged in the flood, and high ground could no longer be distinguished from low. The cows and other animals, shivering from the excessive rain and wind, and the cowherd men and ladies, pained by the cold, all approached Lord Govinda for shelter. Trembling from the distress brought about by the severe rainfall, and trying to cover their heads and calves with their own bodies, the cows approached the lotus feet of the Supreme Personality of Godhead: "Kṛṣṇa, Kṛṣṇa, O most fortunate one, please deliver the cows from the wrath of Indra! O Lord, You are so affectionate to Your devotees. Please save us also."
Seeing the inhabitants of His Gokula rendered practically unconscious by the onslaught of hail and blasting wind, the Supreme Lord Hari understood that this was the work of angry Indra. "Because We have stopped his sacrifice, Indra has caused this unusually fierce, unseasonable rain, together with terrible winds and hail. By My mystic power I will completely counteract this disturbance caused by Indra. Demigods like Indra are proud of their opulence, and out of foolishness they falsely consider themselves the Lord of the universe. I will now destroy such ignorance. Since the demigods are endowed with the mode of goodness, the false pride of considering oneself the Lord should certainly not affect them. When I break the false prestige of those bereft of goodness, My purpose is to bring them relief. I must therefore protect the cowherd community by My transcendental potency, for I am their shelter, I am their master, and indeed they are My own family. After all, I have taken a vow to protect My devotees."
Having said this, Lord Kṛṣṇa, who is Viṣṇu Himself, picked up Govardhana Hill with one hand and held it aloft just as easily as a child holds up a mushroom. "The Lord then addressed the cowherd community: O Mother, O Father, O residents of Vraja, if you wish you may now come under this hill with your cows. You should have no fear that this mountain will fall from My hand. And don’t be afraid of the wind and rain, for your deliverance from these afflictions has already been arranged."
Their minds thus pacified by Lord Kṛṣṇa, they all entered beneath the hill, where they found ample room for themselves and all their cows, wagons, servants and priests, and for all other members of the community as well. Lord Kṛṣṇa, forgetting hunger and thirst and putting aside all considerations of personal pleasure, stood there holding up the hill for seven days as the people of Vraja gazed upon Him. When Indra observed this exhibition of Lord Kṛṣṇa’s mystic power, he became most astonished. Pulled down from his platform of false pride, and his intentions thwarted, he ordered his clouds to desist. Seeing that the fierce wind and rain had now ceased, the sky had become clear of rainclouds, and the sun had risen, Lord Kṛṣṇa, the lifter of Govardhana Hill, spoke to the cowherd community as follows. "My dear cowherd men, please go out with your wives, children and possessions. Give up your fear. The wind and rain have stopped, and the rivers’ high waters have subsided."
After collecting their respective cows and loading their paraphernalia into their wagons, the cowherd men went out. The women, children and elderly persons gradually followed them. While all living creatures looked on, the Supreme Personality of Godhead put down the hill in its original place, just as it had stood before. All the residents of Vṛndāvana were overwhelmed with ecstatic love, and they came forward and greeted Śrī Kṛṣṇa according to their individual relationships with Him — some embracing Him, others bowing down to Him, and so forth. The cowherd women presented water mixed with yogurt and unbroken barleycorns as a token of honor, and they showered auspicious benedictions upon Him.