What you have described is not Buddhism but Hinduism.
I would lke to paste the answer from Seal(GAVE A VERY GOOD ANSWER TO A SIMILAR QUESTION).
Reincarnation, literally "to be made flesh again", is a doctrine or mystical belief that some essential part of a living being (in some variations only human beings) survives death to be reborn in a new body. This essential part is often referred to as the Spirit or Soul, the 'Higher or True Self', 'Divine Spark', 'I' or the 'Ego' (not to be confused with the ego as defined by psychology). According to such beliefs, a new personality is developed during each life in the physical world, but some part of the being remains constantly present throughout these successive lives as well.
Belief in reincarnation is an ancient phenomenon. This doctrine is a central tenet within the majority of Indian religious traditions, such as Hinduism (including Yoga, Vaishnavism, and Shaivism), Jainism, and Sikhism. The idea was also entertained by some Ancient Greek philosophers. Many modern Pagans also believe in reincarnation as do some New Age movements, along with followers of Spiritism, practitioners of certain African traditions, and students of esoteric philosophies.
The Buddhist concept of Rebirth although often referred to as reincarnation differs significantly from the Hindu-based traditions and New Age movements in that the "self" (or soul) does not reincarnate.
Rebirth in Buddhism is the doctrine that the consciousness of a person (as conventionally regarded), upon the death or dissolution of the aggregates (skandhas) which make up that person, becomes one of the contributing causes for the arising of a new group of skandhas which may again be conventionally considered a person or individual. The consciousness arising in the new person is neither identical to, nor different from, the old consciousness, but forms part of a causal continuum or stream with it. The basic cause for this persistent re-arising of personality is the abiding of consciousness in avidya (ignorance); when ignorance is uprooted, rebirth ceases.
Although the cessation of a life is not in itself a sufficient condition for the inception of a new life (since arhats, pratyekabuddhas and buddhas pass away without rebirth), the supporting conditions for a new birth are almost always present. From an external perspective, each life appears as a link in a beginningless sequence of lives, varying in length and in quality.
In traditional Buddhist cosmology, these lives can be in any of a large number of states of being, including those of humans, any kind of animal, and several types of supernatural being (see Six realms). The type of rebirth that arises at the end of one life is conditioned by the karmas (actions of body, speech and mind) of the previous life; good karmas will yield a happier rebirth, bad karmas will produce one which is more unhappy.
In the traditional Buddhist languages of Sanskrit and Pāli, there is no word corresponding exactly to the English "rebirth". A rebirth, that is, the state one is born into, is referred to as jāti, i.e. simply "birth", also referring to the process of being born or coming into the world in any way. The entire process of change from one life to the next is called punarbhava (Sanskrit) or punabbhava (Pāli), literally "becoming again"; it is also known simply as bhava, i.e. "becoming". The process seen from a universal perspective, encompassing all living beings, is called saṃsāra.
From an interior perspective, a person who remembers or imagines a past life is likely to think of it as representing a continuity of existence between lifespans, i.e., that the same person (however defined) was formerly one person (with a certain name and body) and is now a different person (with another name and body). This perspective is objectionable from the point of view of Buddhist philosophy on two counts. First, because it seems to postulate an enduring, self-existing entity that exists separate from the elements of mind and body, contrary to the Buddhist philosophical position of anātman. Second, because it overlooks the characterization of this process as one of constant change, both within and between lives, in which the newly-arising life is conditioned by but in no respect identical to the predecedent life.
Nonetheless, the Buddha is represented using language reflecting the interior perspective in stories about his past lives in both jātakas and sūtras. For instance, "At that time I was the Brahmin, the Great Steward..." (Mahāgovinda-sutta, DN.19) or "Six times, Ānanda, I recall discarding the body in this place, and at the seventh time I discarded it as a wheel-turning monarch..." (Mahāsudassana-sutta, DN.17). This can be regarded as a concession to the needs of conventional speech.