In terms of human society, the idea or meme of the avatara is one that originally developed within the Indian religion of Vaishvanism. In terms of metaphysical realities (assuming avatars etc are valid and real), as a rough definition you could say that the Avatar is a direct emanation and incarnation of the Supreme, of the Absolute, in human (or other limited - e.g. animal) form; although perhaps (or perhaps not) along the lines or channel of a particular God or Godhead.
Ignoring here the problems of historical validity of course; Krishna himself, at least in his current religious folklore form, is obviously a mythological character, a creation of the various writers of the Mahabharata and the Puranas. There may still have been an actual historic Krishna though. The much more humanly defined figure of Jesus makes a better canditate for a historical Avatar, even allowing for all the religious and theological distortion that developed after his death. One could even say that any truely great spiritual teachers were avatars.
It seems a common tendency among modern-day gurus, such as Da Free John (Adi Da), Meher Baba, Sathya Sai Baba, Mataji, and many others, all claim to be avatars, or even in some cases the supreme avatar, of this age, and are each considered the avatar of this age by their respective devotees (how many avatars of this age can there be???). See also Wikipedia for a List of people who have been considered avatars. And while not quite going that far, Swamiji Krishna, who I saw many years ago, claimed to be the sadguru or jagadguru (world-teacher) of this age. Most of these avatar and jagadguru claimants teach the way of Devotion centered on their personality to their followers, in contrast to authentic Realizers who having undergone actual egodeath are very self-effacing.
Adi Da is an interesting example of an avatar claimant who did not originally claim to be the one and only avatar of all times, but simply an enlightened being. In his earlier books, like the The Knee of Listening, The Enlightenment of the Whole Body, and even The Dawn Horse Testament, he presents himself, quite reasonably and persuasively, as an adept - an adept enlightened even from birth, but not as the sole representative of God(hood) on Earth in this era. His claims then became increasingly grandiose; he started saying he was the Kalki avatar, the avatar prophesised in Hindu mythology. I originally thought this was rubbish his stupid followers were putting on him. As stupid humans have done to so many spiritual teachers since time immemorable. But then it transpired that, no, he himself genuinely believes he is the Avatar of this age, and more than that, more recently that he is the only avatar of any age, and teaches his followers to consider him as such. This combination of genuine transcendent experience, and statements that appear quite inflationary, seems more indicative of intermediate zone ensnarement.
In the case of Meher Baba there is a similar claim of exclusiveness. Meher Baba says in former incarnations he was Buddha, Jesus, Mohammad, and so on. Likewise Sathya Sai Baba said he was Krishna who had come again (I think Meher Baba also says he was Krishna), as well as the one who sent Jesus, hence his statement, often cited by Sai Baba devotees that Jesus said "he who sent me will come again". Re this latter, when I first heard it I thought it was actually in the Bible but no it isn't, but it is quoted by SSB devotees as if it is.
There is the problem among of how to reconcile the claims of avatarhood of the guru or master with their human qualities, especially when they are young. The idea seems universally to be that these beings are born avataric but have a sort of veil that conceils the divine quality at first. Then there is usually some sort of touch or awakening.
According to his followers, Meher Baba was an avatar whose human characteristics wree a mere veil until his divinity was unveiled by being kissed on the forehead by Hazrat Babajan.
Similar claims of avataric awakening have been made by other gurus. Sathya Sai Baba supposedly spontaneously realised his avatarhood when he was 12. Before then he was like a normal boy. Da Free John (Adidam) lived a normal life as Franklin Jones, and having attained realisation he never claimed to be an avatar. Afterwards he did (and does) and he and his followers claim he was "born enlightened" but had to go thru the state of normal ignorance before (re)attaining it. This may be based on the Mahayanist position regarding Gautama Buddha - as a bodhisattva he was already enlightened, but had to go through the play of searching for enlightenment etc as a way of teaching by example, a "Skillful Means" (Upaya).
I originally suggested (in March 1999) the term Avatar Stream to disignate the phenomenon of the avataric or godheadic impulse incarnating through a number of different personalities and bodies, each of which continues the same theme.
My inspiration at the time was Sathya Sai Baba, who claimed to have reincarnated in his Satya Sai body after incranating previously in his Shirdi Sai body. However it is now evident that Sai Baba is an abusive guru and hence cannot be an avatar, or even a basic enlightened or liberated being.
A more recent, but equally unpersusasive (at least to the present writer), claimant is Patrizia Norelli-Bachelet, who claims to be the "Third", an avatar of the "solar line", and the successor to Sri Aurobindo and The Mother. She also says that her sone is the "Fourth" and the reincarnation of Sri Aurobindo.
Does this mean that the avatar stream is an invalid concept?
Not necessarily. Just because fakes appropriate an idea doesn't necessarily make it false. The Mother (Mirra Alfassa) refers in passing to a succession of avatars (ref). She also speaks of previous and even simultaneous incarnations.
The idea of a stream or current of spiritual energy is one that is also important in some areas of Hermetic and Magickal thinking. Here there is less emphasis on the personality and more on the occult impulse at work.
Obviously, avataric streams (if they exist) manifest on different levels and planes of existence. There is the physical or incarnate bodies, the subtle or psychic currents working on transformation behind the scenes, still more abstract higher spiritual and noetic impulses, and finally the essence or true heart of the avatar and avataric impulse, and indeed of any impulse to manifestation, which is the Absolute Reality. Particularily the dynamic or manifest Absolute Reality, the mediator between the transcendent infinite eternal and the manifest world of form and change.
While there are many fake avatars, authentic or even half-authentic candidates - assuming the avatar concept is viable - are very rare.
Perhaps the best known contemporarry avatar claimant was Meher Baba, a non-abusive and selfless guru who behaved in an erratic way and claimed to be an avatar. According to his followers, Meher Baba was not merely a God-realized individual, but an avatar -whose vidnyani sanskaras (human characteristics) provided a mere veil until Hazrat Babajan kissed him on the forehead while under her neem tree in Poona, India, circa 1915, thereby unveiling his divinity. He teaches that he was the first soul and in former incarnations he was Buddha (even though Buddha entered nirvana so he was never coming back), Jesus, Mohammad (who always said he was only a prophet, not God) and so on. Compare this with Da's claims. Yet unlike Da, Meher Baba behaved in a selfless way. I have always felt an amazing Light and sincereity about Meher Baba, and beleive him to be genuine. But his over-inflated claims indicate that his outer being may not have been strong enough to hold the higher Light and Force, hence his erratic behaviour.
Better canditates for avatarhood in recent times are Sri Aurobindo and The Mother; the harbingers of the Supramental manifestation; indeed, they would seem to be the only canditates for full avatarhood in recent history. I would suspect that authentic avatars are extremely rare. Christ may have been the next most one, but here there is the question of whether the avataric or "Cosmic Christ" - to use the expression of Rudolf Steiner, Alice Bailey and the early David Spangler - was the same as the historical Jesus or not. Steven Guth has suggested that the same principle that manifested as the Christ consciousness also appeared in the East as Kwan Yin. According to Steiner and the early Spangler, the incarnation in the subtle physical of the Cosmic Christ was a major turning point in the Earth's collective spiritual evolution. Certainly exactly the same could be said, and even more so, today concerning the Supramental manifestation brought by Sri Aurobindo and The Mother.