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Afterlife usually refers to "life after death", "eternal life", or something of the like of the soul. The soul is believed to be the spiritual part of the body that, as stated above, will live forever in the afterlife.
The painful truth is that people sometimes feel deepdown that when they die, they are gone forever. Therefore, they make up things like an afterlife as a form of willful ignorance so that they don't have to think about their ultimate fate.
The trouble is once they have made something up their imagination can run riot. In the end people can imagine an afterlife that is far worse than annihilation. See The Christian concept of Hell. Belief in an afterlife can in cause more pain than atheists experience when they know they will end at death.
Purgatory is the line in between, because there is always something in between.
Hindus and some Buddhists believe that the soul is reincarnated into another animal (incidently, people don't get reincarnated into plants, fungi, bacteria, protists, or abiotic objects) They will believe this cycle will contunue, until you've proben yourself worthy of Nirvana. In this case, you'll lose your memory every time, until Nirvana, and the Hell is earth.
Some people believe in ghosts, which are the souls of people who are on earth, either because they have chosen this way, or they have unfinished business. This means that Earth is Heaven and Hell at the same time.