Reincarnation literally means ‘to be embodied again’, and is the process by which the soul travels from one body to another in successive lives. It is a commonly held belief in many of the world’s spiritual traditions – especially those of the east such as Buddhism, Hinduism, and Taoism.
Reincarnation is elaborately described in Vedic Literature, particularly in the Bhagavad-gita:
As a person puts on new garments, giving up old ones, the soul similarly accepts new material bodies, giving up the old and useless ones. — Bhagavad-gita 2.22
Changing from one body to another is similar to the experience we have even within our present body as we grow up and our body changes from a baby-hood, to a childhood, to a youth, and finally to old age. As these changes happen naturally, so we also pass into another body at death.
It is described that when someone finds tranquility with this process, and comes to realise him or herself to be a spiritual being residing in “the city of the body” then he or she is not bewildered by the transformations of the body and attains real peace.
However, since the soul is eternal then unless this process of reincarnation is stopped it can go on eternally. Therefore, according to the Vaishnava tradition reincarnation is a cycle that needs to be broken. Getting caught up in this cycle of birth, death, and rebirth is not the point. Thus, when the subject of liberation is discussed, it means liberation from this perpetual cycle.
The process of Krishna consciousness is aimed at attaining liberation from the natural difficulties that life brings, and of attaining an eternal spiritual life.
If you would like to learn more about reincarnation there is a very good book on the subject called Coming Back.
As we live through thousands of dreams in our present life, so is our present life only one of many thousands of such lives which we enter from the other more real life, and then return after death. Our life is but one of the dreams of that more real life, and so it is endlessly, until the very last one, the very real, the life of God.” — Count Leo Tolstoy
To fully understand the process of reincarnation, it is essential to also understand karma.