No, reincarnation does not end. It is a cycle you will continue to go through until you reach nirvana, a state of complete enlightenment. Once you reach nirvana, you will no longer be subject to the cycle of rebirth and will be freed from the suffering of this world.
You can aim for nirvana by following the path of Dharma, which is the path of righteousness. Dharma and other spiritual practices can also help you to remember your previous incarnations, so you can learn more about them by speaking to a past life psychic.