What exactly happens after death according to Hinduism?
The simple answer is, we can either be reborn (punar-janma) and experience life once again, or be liberated (moksha) from the cycle of rebirth (samsara). However, the answer is a bit more complex if we see it geographically and historically.
Around the world, what happens after death can be divided into two schools. Those who believe you live only once and those who believe you live multiple lives.
Those who believe you live only once have broadly three schools — those who believe death is the end, nothing else after that; those who believe after death you go to the land of the dead and stay in this afterlife forever; and those who believe after death you go to either heaven, where you enjoy the rest of eternity, or to hell, where you suffer for all eternity (or maybe until you have been adequately punished and are ready to join the rest in heaven).
Those who believe in rebirth believe you keep coming back from the land of the dead (pitr-loka) to the land of the living (bhu-loka) until you learn the ultimate lesson after which you no longer feel the need for a body. There are variations on this, where you are punished for various crimes in hell (naraka-loka) before you are ready to be reborn, or where you enjoy heaven (swarga-loka), until it is time for you return to earth once again.
Ancient Egyptians built pyramids because they believed in an eternal afterlife. Ancient Chinese, before Buddhism introduced the idea of rebirth, have always believed in the land of ancestors that one has to go to after death. Even today, there are rituals where you offer paper money to ancestors to spend in the land of the dead, from whence there is no return.
While re-birth and re-death (punar-mrityu) are seen as inevitable, Hindus have also believed in the concept of immortality (amrita). The devas who live in the sky and the asuras who live under the death fight over this nectar, as do birds (garuda) and snakes (naga). We hear that asuras have Sanjivani Vidya, by which they can resurrect the dead. This is used by Jayanta to bring Shukra back to life. We hear in the Mahabharata, the serpents have naga-mani, or serpent jewel, that can bring back the dead to life; this is used to bring Arjuna back to life after he is shot dead by Babruvahana.
Historically, in the Vedas, we do not find a clear reference to rebirth. There is reference to how our body, after it dies, returns to nature, just like the primordial purusha: so his eye becomes the sun, his breath becomes the wind. There is reference to something that outlives death: atman, jiva, manas, prana. There is reference to a happy land of ancestors and gods (swarga) and to the painful land below the three heavens (naraka). There is reference to feeding the ancestors (pitr). But the idea of rebirth as we know it today is not yet formed.
The idea of rebirth evolves in the Upanishads and fully expressed in the Puranas. While the Vedic householders believed performance of yagna and worldly duties (dharma) took one to heaven, the Vedic hermits spoke of the karma theory, of immortality, of uniting the individual self (atma, jiva-atma) with the cosmic self (brahman, param-atma) through meditation (dhyana), austerities (tapasya) and various social, mental and physical exercise (yoga).
What we find are two options merging: return to this world in another form, or escape to another world. Hence, Hindu rituals are a combination of fire (for escape) and water (for rebirth). There are even communities that choose burial. There are communities that feed ancestors in rituals (shradh) and promise to help their rebirth. In this ritual we focus on relationship of food (anna) and flesh (anna-kosha), and how the dead yearn to return to the land of the living, have a flesh and consume food, while striving for liberation.
Then there is the concept of voluntary renouncing the body (samadhi), which rationalists argue is actually self-termination of life after fulfilling worldly duties. For example, Ram in the Ramayana walks into the river Sarayu and does not rise again after he passes on his kingdom to his children. Likewise, Pandavas walk away into mountains after passing on their kingdom to the next generation. Is this suicide? The faithfuls see it as merging of jiva-atma with param-atma voluntarily by yogis. The skeptics disagree.
Now since suicide is a sin in Christianity, until recently in India, in keeping with its colonial legacy, attempting suicide was a crime. However, Indians have had a mature relationship with death. It is perfectly fine to voluntarily give up life, after completing all worldly duties, with permission of those of family. This is controversial today, but a common theme in the Puranas. Hence, the concept of sanyasa-ashrama, the final stage of life, when you walk away from worldly duties and focus on the divine.
Also read: Can I be a Hindu and still an atheist?