There are different types of incarnations of the Supreme personality of Godhead. In the sastras it is said that Garuda (the carrier of Lord Visnu) and Lord Siva and Ananta are all very powerful incarnations of the Brahman feature of the Lord.
Vinata [Suparna] and her husband Kasyapa gave birth to Garuda, the carrier of Lord Visnu, and to Anuru, or Aruna, the chariot driver of the sun-god. Kasyapa is the son of Marici, is one of the prajapatis, and is one of the sons-in-law of Prajapati Daksa.
On Salmalidvipa Island there is a salmali tree, from which the island takes its name. That tree is as broad and tall as the plaksa tree--in other words 100 yojanas [800 miles] broad and 1,100 yojanas [8,800 miles] tall. This tree is the residence of Garuda. Garuda was given elephants and tortoises as eatables, and is a favored devotee of the Lord.
Garuda is also said to be an inhabitant of the planet known as Kinnaraloka. The inhabitants of the Kinnara planet have the same features as Garuda. Their bodily features are like those of a human being, but they have wings
Once upon a time, Garuda, the carrier of the Lord, snatched away a nectar pot from the hands of the demigods in heaven in order to liberate his mother, Vinata, from the clutches of his stepmother, Kadru, the mother of the serpents. On learning of this, Indra, the King of heaven, hurled his thunderbolt against Garuda. With a view to respect the infallibility of Indra's weapon, Garuda, though otherwise invincible, being the Lord's own mount, dropped one of his wings, which was shattered to pieces by the thunderbolt. The inhabitants of higher planets are so sensible that even in the process of fighting they observe the preliminary rules and regulations of gentleness. In this case, Garuda wanted to show respect for Indra; since he knew that Indra's weapon must destroy something, he offered his wing.
A sparrow laid her eggs on the shore of the ocean, but the big ocean carried away the eggs on its waves. The sparrow became very upset and asked the ocean to return her eggs. The ocean did not even consider her appeal. So the sparrow decided to dry up the ocean. She began to pick out the water in her small beak, and everyone laughed at her for her impossible determination. The news of her activity spread, and at last Garuda, the gigantic bird carrier of Lord Visnu, heard it. He became compassionate toward his small sister bird, and so he came to see the sparrow. Garuda was very pleased by the determination of the small sparrow, and he promised to help. Thus Garuda at once asked the ocean to return her eggs lest he himself take up the work of the sparrow. The ocean was frightened at this, and returned the eggs. Thus the sparrow became happy by the grace of Garuda.
The hymns forming the basis of the Sama Veda were vibrated by the flapping wings of the Lord's carrier, Garuda. In the Vedic literature it is stated that the two wings of the transcendental bird Garuda, who carries the Lord everywhere, are two divisions of the Sama Veda known as brhat and rathantara. Garuda works as the carrier of the Lord; therefore he is considered the transcendental prince of all carriers. With his two wings Garuda began to vibrate the Sama Veda, which is chanted by great sages to pacify the Lord. The Lord is worshiped by Brahma, by Lord Siva, by Garuda and other demigods with selected poems, and great sages worship Him with the hymns of Vedic literatures, such as the Upanisads and Sama Veda. These Sama Veda utterances are automatically heard by the devotee when another great devotee of the Lord, Garuda, flaps his wings.
The Lord descends from His abode, Vaikuntha, in the spiritual sky, and is carried by Garuda. Kardama Muni, who was a liberated soul, saw the Supreme Lord in person, face to face. He could also see Garuda carrying the Lord on the way to Vaikuntha and hear the flapping of his wings vibrating the sound of Hare Krsna, the essence of the Sama Veda.
The snakes who live in the planetary system known as Mahatala are very powerful and have many hoods. They live with their wives and children and consider themselves very happy, although they are always full of anxiety because of Garuda, who comes there to destroy them. This is the way of material life. Even if one lives in the most abominable condition, he still thinks himself happy with his wife, children, friends and relatives.
The Supreme Personality of Godhead, Narayana, seated on Garuda, personally appeared before Gajendra, the King of the elephants. Gajendra, by lifting his trunk, offered obeisances to the Lord, and the Lord immediately pulled him from the water along with the crocodile who had captured his leg. Then the Lord killed the crocodile and thus rescued Gajendra.
In one of his pastimes, the Lord very easily lifted Mandara Mountain with one hand and placed it on the back of Garuda. Then, He too got on the back of Garuda and went to the ocean of milk, surrounded by the demigods and demons. The Lord sat on the mountain and was carried to the spot of the churning by Garuda, who placed the mountain in the middle of the sea. Then the Lord asked Garuda to leave that place because as long as Garuda was present, the snake Vasuki, who was to be used as the rope for churning, could not go there in the presence of Garuda. Garuda, the carrier of Lord Visnu, is not a vegetarian. He eats big snakes. Vasuki, being a great snake, would be natural food for Garuda, the chief of birds. Lord Visnu therefore asked Garuda to leave so that Vasuki could be brought to churn the ocean with Mandara Mountain, which was to be used as the churning rod.
When Caitanya Mahaprabhu appeared in the house of Srivasa Thakura in His Caturbhuja murti, Murari Gupta became His carrier in the form of Garuda, and in these pastimes of ecstasy the Lord then got up on his back.
Once the heavenly king, Indra, was sending torrents of rain over Vrndavana. Garuda was observing the incident from above the clouds, and because of his anger, he began perspiring. When Krsna was fighting with the Kaliya snake by dancing on his heads, Kaliya bit Krsna on the leg. At that time Garuda became infuriated and began to murmur, "Krsna is so powerful that simply by His thundering voice the wives of Kaliya have had miscarriages. Because my Lord has been insulted by this snake, I wish to devour him immediately, but I cannot do so in the presence of my Lord, because He may become angry with me." This is an instance of eagerness to act in ecstatic love as a result of dishonor to Krsna.
The Lord also assured Kaliya: "You came here out of fear of Garuda, who wanted to eat you in the beautiful land by the ocean. Now, after seeing the marks where I have touched your head with My lotus feet, Garuda will not disturb you."
The Lord was fighting on the back of Garuda, and Garuda was also helping the Lord by striking the horses and the elephants with his wings and scratching their heads with his nails and sharp beak. The elephants were feeling much pain by Garuda's attack on them, all were all dispersing from the battlefield. Bhaumasura alone remained on the battlefield, and he engaged himself in fighting with Krsna. He saw that Krsna's carrier, Garuda, was causing great disturbance to his soldiers and elephants, and in great anger he struck Garuda with all his strength, which defied the strength of the thunderbolt. Fortunately, Garuda was not an ordinary bird, and he felt the strokes given by Bhaumasura just as a great elephant feels the impact of a garland of flowers.
Bhaktivedanta Book Trust. Excerpted from various sources, including text and Purport of HDG A.C. Bhaktivedanta Swami Srila Prabhupada.