Avatars of Krishna
The Srimad Bhagavatam 1.3.28 says:
"All of the above-mentioned incarnations are either plenary portions or portions of the plenary portions of the Lord, but Lord Sri Krishna is the original Personality of Godhead. All of them appear on planets whenever there is a disturbance created by the atheists. The Lord incarnates to protect the theists."
Also Lord Brahma, in the first verse of the Bramha Samhita says:
"Krishna, who is known as Govinda is the Supreme Godhead. He has an eternal blissful spiritual body. He is the origin of all. He has no other origin and He is the prime cause of all causes."
Lord Shiva is described in the Brahma-samhita to be like curd or yogurt. Curd is not different from milk. Since milk is transformed into curd, in one sense curd is also milk. Similarly, Shiva is in one sense the Supreme Personality of Godhead, but in another sense he is not, just as curd is milk yet we have to distinguish between the two.
Lord Brahma also explains that many candles lit by a single candle may all look similar, yet one candle remains the original. Krishna is like the original candle, because He is the source of Lord Siva and all other spiritual and material energies.
"...and wherever the Lord is present, the goddess of fortune is also there because the goddess of fortune never lives alone without her husband. Therefore, victory and fortune were awaiting Arjuna, as indicated by the transcendental sound produced by the conch shell of Vishnu, or Lord Krishna." (Bg. 1.14)
"...Persons who are led by the material conception of life do not know that the aim of life is realization of the Absolute Truth, Vishnu, or Bhagavan Sri Krishna, and they are captivated by the external features of the material world, and therefore they do not know what liberation is. .."(Bg. 2.2)
Simultaneously, Srila Prabhupada would speak of Lord Vishnu as that expansion of the Lord Who would take care of business in a more formal way - protecting His devotees, riding on Garuda (Lord Vishnu's great winged carrier), in the heart of each soul - and Lord Krishna as the original form of the Lord who appears in Goloka Vrindavan, plays with the gopas (cowherd boys), and has loving relationships with His dear devotees.
As in the quotes below, from the 9th Chapter of Bhagavad Gita As It Is:
"The mahatma (great soul) does not divert his attention to anything outside Krishna, because he knows perfectly well that Krishna is the original Supreme Person, the cause of all causes. There is no doubt about it. Such a mahatma, or great soul, develops through association with other mahatmas, pure devotees. Pure devotees are not even attracted by Krishna's other features, such as the four-armed Maha-Vishnu. They are simply attracted by the two-armed form of Krishna. They are not attracted to other features of Krishna, nor are they concerned with any form of a demigod or of a human being. They meditate only upon Krishna in Krishna consciousness. They are always engaged in the unswerving service of the Lord in Krishna consciousness." (Bg. 9.13)
. . .or, from the 11th Chapter of Bhagavad-gita:
"This chapter reveals Krishna as the cause of all causes. He is even the cause of the Maha-Vishnu, from whom the material universes emanate. Krishna is not an incarnation; He is the source of all incarnations. That has been completely explained in the last chapter." (Bg. 11.1)
It is summed up by Lord Brahma in his Brahma Samhita as follows:
jivanti loma-vila-ja jagad-anda-nathah
visnur mahan sa iha yasya kala-viseso
govindam adi-purusam tam aham bhajami
"The Maha-Vishnu, into whom all the innumerable universes enter and from whom they come forth again simply by His breathing process, is a plenary expansion of Krishna. Therefore I worship Govinda, Krishna, the cause of all causes." Therefore one should conclusively worship the personal form of Krishna as the Supreme Personality of Godhead who has eternal bliss and knowledge. He is the source of all forms of Vishnu, He is the source of all forms of incarnation, and He is the original Supreme Personality, as confirmed in Bhagavad-gita."(Brahma-samhita 5.48)
Also in Brahma Samhita Lord Krishna is described as the "original candle" from which many candles are lit, and they appear the same, but the original is Lord Sri Krishna!
Krishna Himself says in the Bhagavad-gita that He appears on earth whenever and wherever there's a decline in religious principles and a rise of irreligion. It may seem to us that this is already happening, and we may wonder why He isn't appearing all the time.
This question was also asked at a gathering of sages just after Krishna's departure from the world, around five thousand years ago. This was recounted in the Srimad-Bhagavatam, 1.3.43. The answer is that Krishna exists as the Srimad-Bhagavatam, which tells of His many incarnations and His relationships with His devotees. So, the Srimad Bhagavatam is considered an incarnation of the Lord.
Furthermore, Krishna also appears in His name, specifically as the Hare Krishna mantra, so that in these troubled times, Kali Yuga just by chanting the holy name of God anyone can come to know Krishna. The next scheduled incarnation of Krishna is the Kalki avatar, Who is scheduled to appear in 427,000 years.
This is certainly a popular question. Here's one of the most helpful answers we've found, from Teachings of Lord Chaitanya, Chapter 8:
"Although Vishnu is equal to Krishna, Krishna is the original source. Vishnu is a part, but Krishna is the whole. This is the version given by Vedic literatures. In Brahma-samhita the example is given of an original candle which lights a second candle. Although both candles are of equal power, one is accepted as the original, and the other is said to be kindled from the original. The Vishnu expansion is like the second candle. He is as powerful as Krishna, but the original Vishnu is Krishna. Brahma and Lord Shiva are obedient servants of the Supreme Lord, and the Supreme Lord as Vishnu is an expansion of Krishna."
Krishna and Rama are both vishnu-tattva, in the category of the Supreme Person. But one difference is that Krishna is the origin of all other incarnations of God, and Lord Ramachandra could be said to be one aspect of Krishna's personality. Lord Rama's pastimes are those of a valiant king, and His superhuman activities are those of a great hero.
Krishna also performed incomparable feats in his killing of demons in battle when He was acting as Prince of Dwaraka. But His pastimes with His devotees in Vrindavan show an even more private side of His personality, without any trace of the majesty associated with His pastimes in Dwaraka or those He displayed as Ramachandra.
Krishna's Vrindavan devotees have such pure love for Him that they treat Him as an equal—either as a friend, lover, or son. It is said that in Dwaraka, where Krishna appears as royalty, His majesty overpowers His sweetness. But in Vrindavan, where Krishna appears as a simple cowherd boy, His sweetness overpowers His majesty.
As Ravindra-svarupa dasa (William H. Deadwyler, Ph.D.) mentions in the ISKCON Communications Journal, #2.1, January/June 1994:
"According to Gaudiya Vaishnava theology, God has both a public and a private face. When He manifests his power and majesty (aisvarya), He is known as Narayana and is served in awe and reverence. When He sets aside His lordship, however, and allows his beauty and sweetness (madhurya) to overpower his majesty, He is known as Krishna, the all-attractive. In order to enjoy intimate exchanges of love, Krishna causes His confidential devotees to forget that He is God, so that they may serve Him in a fraternal, parental or conjugal mood."
"All of the above-mentioned incarnations are either plenary portions or portions of the plenary portions of the Lord, but Lord Sri Krishna is the original Personality of Godhead. All of them appear on planets whenever there is a disturbance created by the atheists. The Lord incarnates to protect the theists." (Srimad-Bhagavatam, 1.3.28)
The poet Jayadeva Goswami also wrote a beautiful song about Krishna's ten most prominent incarnations:
O Keshava! O Lord of the universe! O Lord Hari who have assumed the form of a fish! All glories to You! You easily acted as a boat in the form of a giant fish to give protection to the Vedas, which had sunk in the turbulent sea of devastation.
O Keshava! O Lord of the universe! O Lord Hari who have assumed the form of a tortoise! All glories to You! The great Mount Mandara rests upon Your gigantic back, which acts as a pivot for churning the ocean of milk and which has thus been marked with a large circular scarlike depression. In this way Your back has become most glorious.
O Keshava! O Lord of the universe! O Lord Hari who have assumed the form of a boar! All glories to You! The earth, which had sunk to the bottom of the Garbha Ocean, sits fixed upon the tip of Your tusk like a spot upon the moon.
O Keshava! O Lord of the universe! O Lord Hari who have assumed the form of a half-man, half-lion! All glories to You! With the wonderfully sharp nails on Your beautiful lotus hands You have ripped apart the wasplike body of Hiranyakashipu.
O Kesava! O Lord of the universe! O Lord Hari who have assumed the form of a wonderful dwarf brahmana! All glories to You! With Your massive steps You deceive King Bali, and with the Ganges water emanating from the nails of Your lotus feet You deliver all living beings in this world.
O Keshava! O Lord of the universe! O Lord Hari who have assumed the form of Bhrigupati [Parashurama]! All glories to You! You bathe the earth in the rivers of blood from the bodies of the demonic kshatriyas [warriors] You have slain. In this way you wash away the sins of the world and give relief to those who suffer in material existence
O Keshava! O Lord of the universe! O Lord Hari who have assumed the form of Ramachandra! All glories to You! In the battle of Lanka You destroy the ten-headed demon Ravana and distribute his heads as a delightful offering to the presiding deities of the ten directions—a feat long desired by them all, whom that monster had much harassed.
O Keshava! O Lord of the universe! O Lord Hari who have assumed the form of plow-wielding Balarama! All glories to You! On Your brilliant white body You wear garments colored like a fresh rain cloud and the river Yamuna, whom You frighten with the blows of Your plow.
O Keshava! O Lord of the universe! O Lord Hari who have assumed the form of Buddha! All glories to You! Your heart filled with compassion, you decry animal slaughter performed according to the scriptural rules of Vedic sacrifice.
O Keshava! O Lord of the universe! O Lord Hari who have assumed the form of Kalki! All glories to You! Like a comet bearing an indescribably terrifying sword, You annihilate the wicked barbarian kings at the end of Kali-yuga.
O Keshava! O Lord of the universe! O Lord Hari, who have assumed these ten different incarnations! All glories to You! O readers, please hear this most excellent hymn by poet Jayadeva, for it bestows happiness and good fortune and is the best thing in this dark world.
O Lord Krishna, I offer my obeisances unto You, who incarnate in these ten forms. In the form of Matsya You rescue the Vedas, as Kurma You bear Mount Mandara on Your back, as Varaha You lift the earth on Your tusk, as Narasimha You tear open the chest of the demon Hiranyakashipu, as Vamana You trick King Bali, as Parashurama You slay all the wicked ksatriyas, as Ramachandra you defeat Ravana, the son of Pulastya, as Balarama You carry a plow, as Lord Buddha You show compassion toward all, and as Kalki You kill the mlecchas [barbarians]."
He appears here once in a day of Brahma, which means once every thousand Kali-yugas. Sometimes there are differences in His pastimes, but they're generally the same. There are unlimited pastimes going on, even now, on other planets within this universe and in unlimited other universes. The Srimad-Bhagavatam gives us a sample of Krishna's many incarnations and avatars, but it would be impossible to recount them all.
The version of the Srimad-Bhagavatam that's available to human society on this earth contains only eighteen thousand verses. On higher planets, where the lifespan and mental capacity of the inhabitants is far superior to ours, the Bhagavatam has many more verses.
Krishna and Vishnu worshippers are both Vaishnavas , since Krishna and Vishnu are both in the Vishnu category, the Supreme Godhead. The specifics of Their worship may vary, but essentially They are the same person.