The history of Bali Mahārāja and his charity to Vāmanadeva is described in the Eighth Canto of Śrīmad-Bhāgavatam. Bali Mahārāja conquered all the planets of the universe by rightful possession. A king can conquer other kings by strength, and such possession is considered to be rightful. So Bali Mahārāja possessed all the lands of the universe, and he happened to be charitably disposed toward the brāhmaṇas. The Lord therefore pretended to be a beggar brāhmaṇa, and He asked Bali Mahārāja for a measurement of three footsteps of land. The Lord, as the proprietor of everything, could take from Bali Mahārāja all the land he possessed, but he did not do so because Bali Mahārāja possessed all those lands by king's rights. When Bali Mahārāja was asked by Lord Vāmana for such small charity, Bali Mahārāja's spiritual master, namely Śukrācārya, objected to this proposal because he knew thatVāmanadeva was Viṣṇu Himself, pretending to be a beggar. Bali Mahārāja did not agree to abide by the order of his spiritual master when he understood that the beggar was Viṣṇu Himself, and he at once agreed to give Him in charity the land requested. By this agreement Lord Vāmana covered all the lands of the universe with His first two steps and then asked Bali Mahārāja where to place the third step. Bali Mahārāja was very glad to receive the Lord's remaining step upon his head, and thus Bali Mahārāja, instead of losing everything he possessed, was blessed by the Lord's becoming his constant companion and doorman. So, by giving everything to the cause of the Lord, one does not lose anything, but he gains everything that he could never otherwise expect.
(Srimad Bhagavatam-------2:7:17------purport).

The word urukrama means "the one whose activities are glorious." Krama means "step." This word urukrama specifically indicates the Lord's incarnation as Vāmana, who covered the whole universe by immeasurable steps. Lord Viṣṇu is powerful, and His activities are so glorious that He has created the spiritual world by His internal potency and the material world by His external potency. By His all-pervading features He is everywhere present as the Supreme Truth, and in His personal feature He is always present in His transcendental abode of Goloka Vṛndāvana, where He displays His transcendental pastimes in all variegatedness. His activities cannot be compared to anyone else's, and therefore the word urukrama is just applicable to Him only.
According to Sanskrit verbal arrangement, kurvanti refers to doing things for someone else. Therefore, it means that the ātmārāmas render devotional service unto the Lord not for personal interest but for the pleasure of the Lord, Urukrama.
(Srimad Bhagavatam-------1:7:10------purport).

In His Vāmana incarnation He posed Himself as a dwarf brāhmaṇa, but when He took possession of His land, promised by Bali Mahārāja, He expanded His footstep to the top of the universe, extending over thousands and millions of miles. So it was not very difficult for Kṛṣṇa to perform a miracle by extending His bodily feature, but He had no desire to do it because of His deep filial love for His mother, Yaśodā.
(Srimad Bhagavatam-------2:7:27------purport).

The highest scientific advancement of the material scientists is atomic energy. But the material scientist is not able to have an estimation of the particles of atoms contained in the whole universe. But even if one is able to count such atomic particles or is able to roll up the sky like one's bedding, even then one is unable to estimate the extent of the prowess and energy of the Supreme Lord. He is known as Trivikrama because once, in His incarnation of Vāmana, He expanded His leg beyond the highest planetary system, Satyaloka, and reached the neutral state of the modes of nature called the covering of the material world. There are seven layers of material coverings over the material sky, and the Lord could penetrate even those coverings. With His toe He made a hole through which the water of the Causal Ocean filters into the material sky, and the current is known as the sacred Ganges, which purifies the planets of the three worlds. In other words, no one is equal to the transcendentally powerful Viṣṇu. He is omnipotent, and no one is equal to or greater than Him.
(Srimad Bhagavatam-------2:7:40------purport).

Kaśyapa Muni was in the category of the living entities, but he had a transcendental son, Vāmanadeva, who was an incarnation of Viṣṇu. Thus although Lord Viṣṇu is the Supreme Personality of Godhead, He offered His respects to Kaśyapa Muni. Similarly, when Lord Kṛṣṇa was a child He used to offer His respectful obeisances to His mother and father, Nanda and Yaśodā. Also, at the Battle of Kurukṣetra, Lord Kṛṣṇa touched the feet of Mahārāja Yudhiṣṭhira because the King was His elder. It appears, then, that the Personality of Godhead, Lord Śiva and other devotees, in spite of their being situated in exalted positions, instructed by practical example how to offer obeisances to their superiors. Lord Śiva offered his respectful obeisances to Brahmā because Brahmā was his father, just as Kaśyapa Muni was the father of Vāmana.
(Srimad Bhagavatam-------4:6:40------purport).

For a devotee of the Supreme Personality of Godhead, everything is possible because the Lord is also known as Urukrama." Urukrama is a name for LordVāmanadeva, who performed the wonderful act of occupying the three worlds with three footsteps. Lord Vāmanadeva requested three paces of land from Mahārāja Bali, and when Mahārāja Bali agreed to grant them, the Lord immediately covered the entire world with two footsteps, and for His third step He placed His foot upon Bali Mahārāja's head. Śrī Jayadeva Gosvāmī says:
chalayasi vikramaṇe balim adbhuta-vāmana
pada-nakha-nīra-janita-jana-pāvana
keśava dhṛta-vamāna-rūpa jaya jagadīśa hare
"All glories to Lord Keśava, who assumed the form of a dwarf. O Lord of the universe, who takes away everything inauspicious for the devotees! O wonderful Vāmanadeva! You tricked the great demon Bali Mahārāja by Your steps. The water that touched the nails of Your lotus feet when You pierced through the covering of the universe purifies all living entities in the form of the River Ganges."
(Srimad Bhagavatam-------5:1:35------purport).

Lord Viṣṇu once approached Bali Mahārāja while the King was performing a sacrifice. The Lord appeared before him as Trivikrama, or Vāmana, and begged alms from the King in the form of three steps of land. With two steps, Lord Vāmana covered all three planetary systems and pierced the covering of the universe with the toes of His left foot. A few drops of water from the Causal Ocean leaked through this hole and fell on the head of Lord Śiva, where they remained for one thousand millenniums. These drops of water are the sacred Ganges River. It first flows onto the heavenly planets, which are located on the soles of Lord Viṣṇu's feet. The Ganges River is known by many names, such as the Bhāgīrathī and the Jāhnavī. It purifies Dhruvaloka and the planets of the seven sages because both Dhruva and the sages have no other desire than to serve the Lord's lotus feet.
(Srimad Bhagavatam-------5:17------summary).

Bali Mahārāja was so powerful that he fought with Indra and took possession of the three worlds. Indra was certainly very advanced in knowledge, but instead of asking Vāmanadeva for engagement in His service, he used the Lord to beg for material possessions that would lie finished at the end of one age of Manu. An age of Manu, which is the duration of Manu's life, is calculated to last seventy-two yugas. One yuga consists of 4,300,000 years, and therefore the duration of Manu's life is 309,600,000 years. The demigods possess their material opulence only until the end of the life of Manu. Time is insurmountable. The time one is allotted, even if it be millions of years, is quickly gone. The demigods own their material possessions only within the limits of time. Therefore Bali Mahārāja lamented that although Indra was very learned, he did not know how to use his intelligence properly, for instead of asking Vāmanadeva to allow him to engage in His service, Indra used Him to beg Bali Mahārāja for material wealth. Although Indra was learned and his prime minister, Bṛhaspati, was also learned, neither of them begged to be able to render loving service to Lord Vāmanadeva. Therefore Bali Mahārāja lamented for Indra.
(Srimad Bhagavatam-------5:24:24------purport).

The Lord appeared in His original form, with ornaments and weapons in His hands. Although this ever-existing form is not visible in the material world, He nonetheless appeared in this form. Then, in the presence of His father and mother, He assumed the form of Vāmana, a brāhmaṇa-dwarf, a brahmacārī, just like a theatrical actor.
The word naṭaḥ is significant. An actor changes dress to play different parts, but is always the same man. Similarly, as described in the Brahma-saṁhitā (5.33, 39), the Lord assumes many thousands and millions of forms (advaitam acyutam anādim ananta-rūpam ādyaṁ purāṇa-puruṣam). He is always present with innumerable incarnations (rāmādi-mūrtiṣu kalā-niyamena tiṣṭhan nānāvatāram akarod bhuvaneṣu kintu). Nonetheless, although He appears in various incarnations, they are not different from one another. He is the same person, with the same potency, the same eternity and the same spiritual existence, but He can simultaneously assume various forms. When Vāmanadeva appeared from the womb of His mother, He appeared in the form of Nārāyaṇa, with four hands equipped with the necessary symbolic weapons, and then immediately transformed Himself into a brahmacārī (vaṭu).
(Srimad Bhagavatam-------8:18:12------translation and purport).

When Bali Mahārāja, thinking Vāmanadeva to be the son of a brāhmaṇa, told Him to ask for anything He liked, Lord Vāmanadeva praised Hiraṇyakaśipu and Hiraṇyākṣa for their heroic activities, and after thus praising the family in which Bali Mahārāja had been born, He begged the King for three paces of land. Bali Mahārāja agreed to give this land in charity, since this was very insignificant, but Śukrācārya, who could understand that Vāmanadeva was Viṣṇu, the friend of the demigods, forbade Bali Mahārāja to give this land. Śukrācārya advised Bali Mahārāja to withdraw his promise. He explained that in subduing others, in joking, in responding to danger, in acting for the welfare of others, and so on, one could refuse to fulfill one's promise, and there would be no fault. By this philosophy, Śukrācārya tried to dissuade Bali Mahārāja from giving land to Lord Vāmanadeva.
(Srimad Bhagavatam-------8:19------summary).

A brāhmaṇa or sannyāsī is qualified to ask charity from others, but if he takes more than necessary he is punishable. No one can use more of the Supreme Lord's property than necessary. Lord Vāmanadeva indirectly indicated to Bali Mahārāja that he was occupying more land than he needed. In the material world, all distresses are due to extravagance. One acquires money extravagantly and also spends it extravagantly. Such activities are sinful. All property belongs to the Supreme Personality of Godhead, and all living beings, who are sons of the Supreme Lord, have the right to use the Supreme Father's property, but one cannot take more than necessary. This principle should especially be followed by brāhmaṇas and sannyāsīs who live at the cost of others. Thus Vāmanadeva was an ideal beggar, for He asked only three paces of land.
(Srimad Bhagavatam-------8:19:17------purport).

The Supreme Personality of Godhead, being full in Himself, actually has nothing to want for His self-interest. Lord Vāmanadeva, therefore, had not gone to Bali Mahārāja for His own self-interest. As stated in Bhagavad-gītā (5.29), bhoktāraṁ yajña-tapasāṁ sarva-loka-maheśvaram. The Lord is the proprietor of all planets, in both the material and spiritual worlds. Why should He be in want of land? Bali Mahārāja rightly said that Lord Vāmanadeva was not at all prudent in regard to His own personal interests. Lord Vāmanadeva had approached Bali not for His personal welfare but for the welfare of His devotees.
(Srimad Bhagavatam-------8:19:18------purport).

Vāmanadeva will first occupy the three worlds with one step, then He will take His second step and occupy everything in outer space, and then He will expand His universal body to occupy everything. Where will you offer Him the third step?
Śukrācārya wanted to tell Bali Mahārāja how he would be cheated by Lord Vāmana. "You have promised three steps," he said. "But with only two steps, all your possessions will be finished. How then will you give Him a place for His third step?" Śukrācārya did not know how the Lord protects His devotee. The devotee must risk everything in his possession for the service of the Lord, but he is always protected and never defeated. By materialistic calculations, Śukrācārya thought that Bali Mahārāja would under no circumstances be able to keep his promise to the brahmacārī, Lord Vāmanadeva.
(Srimad Bhagavatam-------8:19:34------translation and purport).

Bali Mahārāja thought that even if this brahmacārī, Vāmanadeva, were Lord Viṣṇu, if the Lord accepted his charity and then again arrested him, Bali Mahārāja would not envy Him. Considering all these points, Bali Mahārāja finally gave in charity everything he possessed.
Lord Vāmanadeva then immediately extended Himself into a universal body. By the mercy of Lord Vāmanadeva, Bali Mahārāja could see that the Lord is all-pervading and that everything rests in His body. Bali Mahārāja could see Lord Vāmanadeva as the supreme Viṣṇu, wearing a helmet, yellow garments, the mark of Śrīvatsa, the Kaustubha jewel, a flower garland, and ornaments decorating His entire body.
(Srimad Bhagavatam-------8:20------summary).