Reincarnation is the process of rebirth in which a person's spiritual essence, or soul, moves from one body to another in a continuous cycle of birth and death.
In the Bhagavad-gita, this process is described as follows: “Just as the embodied soul continuously passes in this body from childhood to youth and from youth to old age, so at the time of death it passes into another body. A wise person is not bewildered by such a change.” (Bhagavad-gita 2.13)
Because the soul is eternal, this process of transmigration of the soul also continues eternally—unless it is brought to an end.
Our daily lives inevitably contain a certain amount of sorrow and suffering. Since the soul is immortal, it would continue to experience suffering indefinitely as long as it remains within the cycle of reincarnation. The Hare Krishna philosophy explains how one can put an end to this vicious cycle of suffering.
An example of this teaching can be found in the Bhagavad-gita, which is essentially an explanation of how to break free from this cycle and attain liberation. You can read more about this in the section on Karma.