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

The King Milinda once asked the Buddhist sage Nagasena: “When someone is reborn, is he the same as the one who just died, or is he different?”
Nagasena replied: “He is neither the same nor different. . . . Tell me, if a man were to light a lamp, could it provide light the whole night long?”
“Yes.”
“Is the flame then which burns in the first watch of the night the same as the one that burns in the second . . . or the last?”
“No.”
“Does that mean there is one lamp in the first watch of the night, another in the second, and another in the third?”
“No, it’s because of that one lamp that the light shines all night.”
“Rebirth is much the same: One phenomenon arises and another stops, simultaneously. So the first act of consciousness in the new existence is neither the same as the last act of consciousness in the previous existence, nor is it different.”

Sometimes I tease people and ask: “What makes you so adamant that there’s no life after death? What proof do you have? What if you found there was a life after this one, having died denying its existence?”
Those of us who undertake a spiritual discipline—of meditation, for example—come to discover many things about our own minds that we did not know before. For as our minds open more and more to the extraordinary, vast, and hitherto unsuspected existence of the nature of mind, we begin to glimpse a completely different dimension, one in which all of our assumptions about our identity and reality, which we thought we knew so well, start to dissolve, and in which the possibility of lives other than this one becomes at least likely. We begin to understand that everything we are being told by the masters about life and death, and life after death, is real.

The belief in reincarnation shows us that there is some kind of ultimate justice or goodness in the universe. It is that goodness that we are all trying to uncover and to free. Whenever we act positively, we move toward it; whenever we act negatively, we obscure and inhibit it. And whenever we cannot express it in our lives and actions, we feel miserable and frustrated.

“If we have lived before,” I’m often asked, “why don’t we remember it?” But why should the fact that we cannot remember our past lives mean that we have never lived before? After all, experiences—of our childhood, or of yesterday, or even of what we were thinking an hour ago—were vivid as they occurred, but the memory of them has almost totally eroded, as though they had never taken place. If we cannot remember what we were doing or thinking last Monday, how on earth do we imagine it would be easy, or normal, to remember what we were doing in a previous lifetime?

If you were to draw one essential message from the fact of reincarnation, it would be: Develop a good heart that longs for other beings to find lasting happiness, and acts to secure that happiness. Nourish and practice kindness.
The Dalai Lama has said: “There is no need for temples; no need for complicated philosophy. Our own brain, our own heart is our temple; my philosophy is kindness.”

Six realms of existence are identified in Buddhism: gods, demigods, humans, animals, hungry ghosts, and hells. They are each the result of one of the six main negative emotions: pride, jealousy, desire, ignorance, greed, and anger.
Looking at the world around us, and into our own minds, we can see that the six realms definitely do exist. They exist in the way we unconsciously allow our negative emotions to project and crystallize entire realms around us, and to define the style, form, flavor, and context of our life in those realms. And they exist also inwardly as the different seeds and tendencies of the various negative emotions within our psychophysical system, always ready to germinate and grow, depending on what influences them and how we choose to live.

It has often intrigued me how some Buddhist masters I know ask one simple question of people who approach them for teaching: “Do you believe in a life after this one?” They are not being asked whether they believe in it as a philosophical proposition but whether they feel it deeply in their hearts. The master knows that if a man believes in a life after this one, his whole outlook on life will be different, and he will have a distinct sense of personal responsibility and morality. What the masters must suspect is that there is a danger that people who have no strong belief in a life after this one will create a society fixated on short-term results, without much thought for the consequences of their actions.
Could this be the major reason why we have created a world like the one we are now living in, a world with hardly any real compassion?