In simple terms of Hindu Philosophy, every single person on this earth is a unique soul (Jeeva). In this soul, all the database of life's learning & actions go on a permanent basis & all this database remains always in soul in dormant state. There is not only the database of this life but also of previous lives in this soul (Jeeva). At the death of a person, one person's memory and knowledge remains in dormant state in soul & temporarily suppressed while that soul takes a new birth in a new body. So whenever, a person is born, he/she is always carrying the database of his/her previous lives with them. Though most of the time it remains in dormant state but is always reflected as our likes and dislikes, moods & skills. That is the reason that you may find a child prodigy who suddenly excels in something without any experience. In all such cases past lives memories & skills play a major part.
Though modern science do not believe in soul but it is a matter of individual faith as it is endorsed by many religions & Hindu religion is one of them. As per Bhagwat Gita it is told that individual soul is immortal & everlasting; though human body dies, soul does not. It changes the body when the old body dies. It just does that like the way we change our clothes for a new pair, when the old one's are torn. Below are verses of Bhagwat Gita in which Lord Sri Krishna is telling his disciple Arjuna, the mystery of soul & it's rebirth:
"dehino-asmin yatha dehe
kaumaram yauvnam jara
adhiras tatra na muhyati" (Bhagwat Gita: Chapter Two verse 13)
"Sri Krishna said: O Arjuna, Just as boyhood, youth & old age are attributed to the soul & the embodied soul continuously passes through these cycles, similarly the embodied soul passes into another body at death. The wise man does not get deluded & bewildered with such a change."
"na jayate mriyate va kadacin
nayam bhutva bhavita va na bhuyah
ajo nityah sasvato ’yam purano
na hanyate hanyamane sarire" (Bhagwat Gita: Chapter Two verse 20)
navani grhnati naro ’parani
tatha sarirani vihaya jirnany
anyani samyati navani dehi" (Bhagwat Gita: Chapter Two verse 22)
"nainam chindanti shastrani
nainam dahati pavakah
na chainam kledayanty apo
na sosayati marutah" (Bhagwat Gita: Chapter Two verse 23)
"Sri Krishna said: O Arjuna, The soul can never be cut to pieces by any weapon, nor burned by fire, nor moistened by water, nor withered by the wind."
akledyo-asoshya ev ch
nity sarv-gatah sthanur
achalo-ayam sanatanah” (Bhagwat Gita: Chapter II verse 24)
"Sri Krishna said: O Arjuna, For this individual soul is incapable of being cut; and insoluble, and can be neither burned nor dried. This soul is eternal, constant, omnipresent, unchangeable, immovable and everlasting."
"sariram yad avapnoti
yac chapy utkramatishvarah
vayur gandhan ivasayat" (Bhagwat Gita: Chapter Fifteen verse 8)
"Sri Krishna said: O Arjuna, The living entity (individual human soul) in the material world carries his different conceptions of life (individual tastes & actions/karmas) from one body to another as the air carries aromas from one place to other. Thus he takes one kind of body and again quits it to take another."
"srotram chaksuh sparshanam cha
rasanam ghranam eva cha
adhisthaya manas chayam
visayan upasevate" (Bhagwat Gita: Chapter Fifteen verse 9)
"Sri Krishna said: O Arjuna, The living entity (individual human soul), thus taking another gross body, obtains a certain type of ear, eye, tongue, nose and sense of touch, which are grouped about the mind. He thus enjoys a particular set of sense objects through the another body."
"utkramantam sthitam vapi
bhunjanam va gunanvitam
pasyanti jnana-chaksusah" (Bhagwat Gita: Chapter Fifteen verse 10)
"Sri Krishna said: O Arjuna, The foolish cannot understand how a living entity (individual human soul) can quit his/her body, nor can they understand what sort of body he enjoys under the spell of the modes of nature. But one whose eyes are trained in knowledge of soul can see all this."
"yatanto yoginas chainam
pasyanty atmany avasthitam
yatanto ’py akritatmano
nainam pasyanty acetasah" (Bhagwat Gita: Chapter Fifteen verse 11)
"Sri Krishna said: O Arjuna, The endeavoring transcendentalists, who are situated in self-realization, can see all this (soul transmigration) very clearly. But those whose minds are not developed in this knowledge and who are not situated in self-realization cannot see what is taking place, though they may try to."