Reincarnation is the transmigration of soul from one body to another. The soul is residing in a particular body due to its karma or past actions. When that karma is over then the soul leaves the body and is given another body based on new karma under the supervision of the Supreme Lord, this change of body is called reincarnation or in general terms called death. Bhagavad-gita (2.23) compares this to change of dress.
vasamsi jirnani yatha vihaya
navani grhnati naro 'parani
tatha sarirani vihaya jirnany
anyani samyati navani dehi
"As a person puts on new garments, giving up old ones, the soul similarly accepts new material bodies, giving up the old and useless ones." Rebirth is indeed a fact, but as the soul is so minute that we can’t even see it so how can we monitor its movements from one body to another. So we have to take it from the shastra that the soul travels from one life to another. This soul never dies even though the body dies.