The basis on which Buddhist[s] accept the concept of rebirth is principally the continuity of consciousness . . . If you trace our present mind or consciousness back, then you will find that you are tracing the origin of the continuity of mind into an infinite dimension; it is, as you will see, beginningless.
Therefore there must be successive rebirths that allow that continuum of mind to be there.

We are born and reborn countless number of times, and it is possible that each being has been our parent at one time or another. Therefore, it is likely that all beings in this universe have familial connections.

From a Buddhist point of view, the actual experience of death is very important. Although how or where we will be reborn is generally dependent on karmic forces, our state of mind at the time of death can influence the quality of our next rebirth. So at the moment of death, in spite of the great variety of karmas we have accumulated, if we make a special effort to generate a virtuous state of mind, we may strengthen and activate a virtuous karma, and so bring about a happy rebirth.

Ordinarily, it is difficult to remember one's past life. Such recollections seem to be more vivid when the child is very young, such as two or three, and in some cases even younger. ...When the present body is fully formed, the ability to recall past life seems to diminish.
The mental associations with this life become increasingly dominant. There is a close relationship during the first few years of one's life with the continuum of consciousness from the previous life. But as experiences of this life become more developed and elaborate, they dominate.
It is also possible within this lifetime to enhance the power of the mind, enabling one to reaccess memories from previous lives. Such recollection tends to be more accessible during meditative experiences in the dream state. Once one has accessed memories of previous lives in the dream state, one gradually recalls them in the waking state.
From Consciousness at the Crossroads: Conversations with The Dalai Lama on Brain Science and Buddhism