A basic tenet of Hinduism is that we each have a spark of Brahman (or Paraatman-beyond atman) in us, and that only when we fully understand this can our atman (soul) be united with Brahman, thereby attaining Moksha (liberation) from the endless cycle of Samsara (rebirth). However, within Hinduism there are many different theories or beliefs regarding the nature of the relationship between Brahman and atman.
Firstly, what are Brahman and atman exactly? Brahman is the Ultimate Reality, the indescribable It, the divine soul of all existence whose breath penetrates everything, both living and non-living. The best explanation of Brahman is that it is "neti-neti" ("not this, not that"). It manifests itself through the Trimurti, much like the Christian concept of God being manifested in the Trinity. Atman is commonly defined as the soul of all living creatures- Sat C(h)it Ananda that is indestructible and immortal. Hidden within each of us is an all-knowing, ever-blissful Atman that is no less than the infinite Brahman. Once we die, it continues in a new body until Moksha from Samsara. Some believe that atman will return to Brahman as a drop of water to the ocean, while others believe that the atman will retain its individuality but will live in the presence of Brahman. Still others believe that after liberation, we become greater than Brahman.
"Let me tell you about the powers of Vishnu, the God of Light... May this hymn of praise rise up and give strength to Vishnu" (Rig-Veda, I.154: 1-3).
In the Rig-Veda, which is the oldest scripture, we see the idea of humanity sustaining the divine through worship in a unique two-way relationship. The supreme force is seen as that which protects, sustains, and knows all, but because a part of that force is in each of us, humanity is just as indispensable. This idea would seem sacrilegious in most of the other mainstream religions as it assumes equality with the divine.
"You do not perceive that finest essence, the Universal Soul, yet it is here. That is Reality. That is atman, and you are also that essence" (Chhandogya Upanishad, VI.13: 1-3).
In the Upanishads (see Vedanta and Vedas) which were written in c.600 BC, the transcendent realization of tat tvam asi was introduced, with their more introspective and interpretive approach to religion. The relationship between Brahman and the world of maya was developed- that the eternal underlying reality of Brahman can exist within maya, though maya can be a distracting force as our atman tries to reunite with Brahman.
In the famous epic poems Ramayana and Mahabharata, which were written from about 400 BC-400AD, the Supreme Lord is manifested in Rama, the embodiment of Dharma, who comes to the world to restore moral order. Dharma (root dhr=to sustain) is the fluid, broad concept of organizing our lives so that we know our duty.
"Dharma is so called because of its capacity to sustain the world. On account of dharma, people are sustained separately in their respective station" (Mahabharata 12.110.10-11).
Our lives are separated into 4 different stages (Ashramas)- student, householder, forest-dweller, and then ascetic. Before this organization comes our varna/caste that is predetermined by birth. For each caste and for each stage of life, the divine plays a different part in one's life. For example, if one is in any of the top three varnas and a male, they have access to the Vedas, which makes their spiritual relationship with Brahman more intellectual or philosophical, and less simple and direct like the basic devotion to an ishwara (personal god) that someone in the last Varna might worship.
Some philosophers in the 11th century argued that Brahman controls the atman and that God/Brahman/Allah is manifested in the world through our sensory experiences. Therefore, only by worshipping It with selfless devotion can one attain the highest state of bliss, as Brahman needs to be loved by that which is not Itself. In addition, in Moksha the individual retains its self-consciousness. This is called qualified non-dualism. Dualism (Dvaita) is the least common philosophy among Hindus, as it has a strong Christian influence. For example, in the 13th century the philosopher Madhva taught that Moksha is not becoming one with Brahman, but dwelling near It in "eternal contemplation of His glory." | 
