Ancient Sanskrit scripture defines avatar as the embodiment of a divine being on earth. It can be either the descent of a deity like Vishnu or any divine spirit coming down to the earth for establishing righteousness and ordaining evil. The practice of divine incarnation is one of the most in-depth theological ideas of Hinduism.

Understanding Divine Incarnation

In Hinduism, God is both formless (meaning nirguna brahman), or with form (meaning saguna brahman). With avatars, the formless divine consciously takes on a corporeal manifestation for the fulfillment of some cosmic purposes. This is an idea that directly links the transcendent with the material world.

The Bhagavad Gita, one of the most important texts of Hinduism, provides the philosophical basis for incarnations in Chapter 4, verses 7-8:

"Whenever dharma declines and adharma prevails, I manifest myself. To protect the virtuous, to destroy the wicked, and to reestablish dharma, I appear millennium after millennium." (Bhagavad Gita 4:7-8)

The Ten Primary Avatars of Vishnu 

While Hinduism proposes a plethora of avatars, the best-known are those of Vishnu's ten avatars, called Dashavatara:

1. Matsya(Fish): Rescued Manu (progenitor of mankind) from the Great Flood as told in the Matsya Purana and Shatapatha Brahmana 1.8.1. 

2. Kurma(Tortoise): Supported Mount Mandara during the churning of the ocean as detailed in Bhagavata Purana 8.7-8. 

3. Varaha(Boar): Rescued the Earth from the cosmic water as elaborated in Varaha Purana and Vishnu Purana 1.4. 

4. Narasimha(Man-Lion): Slays Hiranyakashipu, the demon king, as depicted in Bhagavata Purana 7.8. 

5. Vamana(Dwarf): Forced Bali the demon king to his knees, narrated in Vamana Purana and Bhagavata Purana 8.18-22. 

6. Parashurama( Warrior with Axe): Destroyed the corrupt warrior clans as described in the Brahmanda Purana and Mahabharata. 

7. Rama: The epitome of righteousness who triumphed over Ravana, celebrated in the Ramayana and Ramacharitamanas. 

8. Krishna: The avatar who narrated the Bhagavad Gita and led the Mahabharata war, tremendously written in the Bhagavata Purana and Harivamsa. 

9. Buddha: In later Hindu traditions, he is recognized as an avatar and an initiator of religious reforms, mentioned in Bhagavata Purana 1.3.24. 

10. Kalki: The projected avatar for the terminal point of the present age (Kali Yuga), foretold in the Kalki Purana and Vishnu Purana 4.24.

Spiritual Significance for the Practitioners
For those who start to learn about and practice Hinduism, the avatar concept provides many spiritual insights:

1. The Divine Accessibility: The transcendent divine through avatars becomes accessible to human's experience and devotion.
2. An Ethical Framework: The stories of each avatar teach moral lessons about proper living and dharma.
3. A Mode of Worship: Bhakti traditions stress a personal relationship with particular incarnations: notably Krishna and Rama.
4. Symbolic of Evolution: A few modern interpreters see the succession of avatars as symbolizing evolutionary progress for humanity.

The Upanishads (Chandogya Upanishad 6.2.1) teach "Tat Tvam Asi" (That Thou Art), implying a deep connection between human consciousness and divine reality. Avatar serves as a bridge connecting this philosophical concept to divine personalities with which devotees can connect and worship.

Modern Application

Modern practitioners of avatar worship commonly find themselves engaged in temple rituals, festival celebrations, and personal devotion (puja). The Bhagavata tradition, however, recommends meditating on the avatars' qualities and stories in the name of spiritual transformation.

As per Padma Purana (6.226.42-44) it says: "By constant remembrance of the Lord's divine play and activities, one becomes free from all sins." An understanding of avatars can constitute an entry point into Hindu theology that, on one hand, attempts to balance those abstract philosophical definitions with the devotional practice clearly implementable in the spiritual life of adherents.