Reincarnation
Etymology
"Incarnation" is from the Latin word, carnis, which means "meat or flesh." To be incarnated is to be embodied, enfleshed.
Hinduism
Many Eastern and New Age religions share the Hindu view of reincarnation, namely that the very same principle of life, jiva, can animate ("bring to life") any imaginable form of life. The same jiva could be incarnated as an insect, a worm, a monkey, an elephant, a horse, a man, a woman, or any other kind of living being.
Hindus believe (as a general rule — there is no Pope, creed, dogma, or catechism for all Hindus) that we will continue to be reincarnated in this universe until we become divine. If we live our life well, the law of karma will reward us by causing us to be reincarnated in a superior form of life; if we live our life poorly, the law of karma will punish us by causing us to be reincarnated in an inferior form of life.
The appeal of this idea is obvious. It takes the pressure off of us for any one lifetime. We don't have to worry about ourselves or those whom we love. If we don't get it right in this lifetime, there are infinitely many more lifetimes available for us to learn our lessons and reach moksha, liberation from the cycle of reincarnation.
The movie, "Groundhog Day," was written to illustrate the doctrines of reincarnation and karma.
The Catholic Understanding of Life and Death
The Soul is the Form of the Body
For Catholics, the soul is the form of the body.
We should not think of the soul as a passenger living inside of a living body, as the Hindus do.
The Catholic doctrine rules out the idea that a soul could be found in various forms (an animal, a male, or a female). The soul itself is a form, a very specific form, of one individual human being. That form can inform only one living being: that particular human person.
We go through life on earth just once
"Death is the end of man's earthly pilgrimage, of the time of grace and mercy which God offers him so as to work out his earthly life in keeping with the divine plan, and to decide his ultimate destiny. When 'the single course of our earthly life' is completed (LG 48 § 3), we shall not return to other earthly lives: 'It is appointed for men to die once' (Heb 9:27). There is no 'reincarnation' after death" (CCC, 1013).