If anyone believes it I am interested to see how they understand it and think about it, but to me i do not get it
My family believe in past lives. Do I? probably not. An interesting plot device for fiction, nonetheless.
But to your question. In the western sense, my family believes something along the lines that People are reborn because they need to learn various lessons. Their idea of Rebirth/Reincarnation is particularly devoid of Hinduism or Buddism or any Dharmic spiritual idea in that they don't believe that Karma determines how one is reborn. Although, they do believe that Karma exists, but rather that people experience the noxious effects of Karma within this lifetime, not the next. The idea of Reincarnation is nice for some people, because it gives one a chance to redeem themself, whereas in other religious systems if one is a POS (Piece of Sh*t) in this lifetime, their soul isn't necessarily condemned to eternal punishment, but has countless chances to be better in other lifetimes and eventually become perfected and merge with the all if their belief goes that far. As an aside, it should be noted that my family trends towards syncretism (attempts at blending religious traditions) so its really not a big deal for them to believe in Jesus and reincarnation and a myriad of other things...
Edit: Since everyone is talking about afterlife in conjunction with past lives, I may as well give my thoughts. It has never been conceivable for to believe that when you die thats it; I have always had a belief in some form of afterlife, but not a traditional belief. Many believe in an afterlife that mirrors the current one we are living, but without the current problems and distortions of life.
There is a scene in BTVS (Buffy the Vampire Slayer) after Buffy is brought back where she describes heaven: "Nothing had form, but I was still me. I knew that everyone I loved was okay. I was warm and I was loved. I was finished." This seems like something similar to my perception of an afterlife. Perhaps, my idea of paradise would mirror an Eastern Orthodox in that Heaven/Hell are states of being where those who love/loved God feel safe and secure and happy at the presence of the divine, whereas those who don't feel uncomfortable or depressed at the love of the divine. Dostoevsky said: What is Hell? I maintain that hell is the inability to love." Take all these random bits I have provided and jumble them into a jigsaw puzzle and you'll get my idea of an afterlife. I'll leave the OP and others to determine what that might be.