The theory of Avatar(Incarnation) is quite old. It is described in details in Srimad Bhagwat, Srimad Bhagavad Gita. Also described in Sri Ramakrishna Kathamrita and book Bhakti Yoga (by Swami Vivekananda).
Avatar is for devotee’s. Does not apply for Jnani’s. Jnani (Individual with supreme knowledge) understand everything is Brahman, so also Avatar. For him
“Purnamadah Purnamidam Purnat purnamudachyate. Purnasya Purnamadaya Purnamevavashisyate ”. So there is no concept of Avatar in Veda and Upanisad. Upanisad say’s each individual is divine(Brahman) by nature.
Each individual is Avatar if you think of Jiva is also incarnating (taking birth) based on it’s past Karmas.
Whereas from this phenomenal world point of view where individual Jiva exists as body and mind conscious (deprived from the knowledge of Divinity), Avatar concept is applied and understood well. Both Jiva and Avatar looks alike from physical appearance. But few differences between Jiva and Avatar are:
1. Jiva incarnates(takes birth) based in its past Karmas (Prarabdha). Whereas Avatar incarnates based on need (to establish Dharma, for entertainment of Devotees etc), called Samasti Prarabdha.
2. Jiva lives to fulfill individual desires and needs, whereas Avatar scarifies whole life for welfare of the world. Their love is for whole human being (not limited to any group, geometrical region like country India etc).
3. An ordinary spiritual Guru (illuminated Jiva) propagates spiritual power through Ista Mantra. Whereas Avatar donates spiritual power through mere wish. – For example on Kalpataru day of Sri Ramakrishna, many got spiritual vision just by grace. Same applies for other incarnation like Krishna (to Arjuna), Rama (to Ahalya) etc.
4. An ordinary spiritual Guru gets illumination and initiates the disciples on his own realized path. Once the Guru departs the physical body, his followed path become weak and does not remain as a strong spiritual current for future generation. Whereas Avatar comes with his own group (already illuminated disciples, here like Swami Vivekananda along with other fifteen direct disciples, Hanuman, Bharata etc came with Sri Rama, Gopis, Arjuna etc came with Krishna). The dharma an Avatar establishes has very strong power and future generations also get benefited even after the Avatar has left this mortal plane. For example what Krishna preached through Gita, Rama preached through his life of truth are still influencing us. Similarly Sri Ramakrishna is a bigger current than all previous Avatar. Becasuse only Sattwa Guna was prevalent in this Avatar which is possible only in Satya Yoga.
5. An Avatar has his Brahman awareness (as well as his past and future life) from his childhood. They know that they are not mere body and mind. So they can throw their body based on their choice (or when the work got completed). Whereas Jiva has only awareness of body and mind consciousness. Jiva needs to do spiritual practice to have Brahman awareness (called self realization).
6. There are so many Avatars incarnated in India. We don’t worship all of them. We worship the Avatar who is incarnated with his divine consort (Shakti) like Krishna with Radha, Rama with Sita, Ramakrishna with Sarada etc. Whereas Parsurama, Kapila muni, Vyasa deva were also Avatar and we don’t worship them because they are incomplete (without any Shakti)
7. Above all an Avatar (Sri Ramakrishna) is our best friend, as he only tells us Jiva is Brahman and through his grace we can have our self realization very easily. He is the best friend, best Guru, best parent and what not. He doesn’t expect anything from Jiva, but gives spiritual illumination free of cost after removing all pollutions from our mind. In return what we can offer him. We can only keep him in our heart and worship him.