The first important observation is that Buddhism is
Buddhism introduces the concept of rebirth. Nirvana and rebirth are interconnected. Rebirth is the equivalent of suffering. Nirvana is the equivalent of the end of suffering, which is the goal of Buddhism. Buddhism starts with few axioms (assumptions) called "The Four Noble Truths". They are called "Noble", because they make the person who understands them a Noble person (at least that's what is claimed).
The first Noble Truth is that "Life means Suffering": Buddha based his visions on the indisputable(?) fact that any person who lives has to suffer one way or another at some point. The second Noble Truth is that "Suffering is caused by Desire": Buddha elaborates on the first truth and tries to explain why people suffer. Desiring transient (earthly) things causes suffering. So if someone wants to get something, but they don't get it, they will feel the suffering. Similarly, the process of acquiring things includes suffering. To get something, you have to pay something... Finally, it is by human nature that we have a never ending desire, that is whenever we achieve one thing, we will start looking for the next thing!! This concludes that life is a series of endless suffering. That's why Buddha has said: "Contentment is the greatest wealth".
The third Noble Truth claims that ending of suffering is possible by eliminating its cause (ie. releasing oneself from seeking earthly things). This axiom is simply conforming to the cause and effect philosophy. The fourth Noble Truth, is that to be able to release yourself from suffering, you can follow "The Eightfold Path", which is a path to the Nirvana.
Buddhism states that desire to live causes a person to live again through a process called "Rebirth". Since living, by the Buddhist definition, is equivalent to suffering, then being reborn means the continuation of suffering. Rebirth can be viewed as transferring the mental energy from a deceased body, to a newly formed living form (fertilized eggs in the case of humans). It is believed that a human can be reborn as a different species on earth, species in distant planets in this galaxy, or even a completely different dimension than this universe. But when a person eliminates his desire to live, he will not be reborn, and thus reaching a state of inexistence... Such state is free from suffering, thus is considered as the goal of the Buddhist. This state is called Nirvana.
Edit: Since some variations of Buddhism accept theism, the general statement "Buddhism is atheist" cannot hold. Although it can be noted that the most spread variations of Buddhism are atheist, particularly Theravada variation which is the oldest and most conforming to the original teachings of The Buddha.