There is no commonly agreed upon definition of
atheist. Just as there are thousands of definitions of "god", there are many (probably not thousands, though) types of atheists. The word, literally means "non-theist" or "anti-theist". Depending on the context, it can range from an active and vocal denial of god and rejection of religions to a simple disinterest in the whole topic. It can imply an active disbelief in deities or simple lack of a belief in them due to their improbability and lack of evidence. Atheists claim there are two main reasons for their denial of the existence of gods and/or disbelief in gods: "Strong Atheists" claim that there is positive evidence that gods do not exist (also called "positive atheism"), while "Weak Atheists" claim that theists bear the burden of proof to show that god(s) exist, that they have failed to do so, and that belief is therefore unwarranted.
Some atheists are casually non-theistic about the dominant god or gods of their culture in the same sense as we all are about Thor, Mithra, Zeus, Marduk, or Isis. They are just apathetic about gods, without distinction. Others (the "new atheists") take an activist stance against religion, asserting they have proof that god doesn't exist.
The word, "atheist" has strong negative connotations. Many atheists are not comfortable identifying publicly as atheists because of that stigma. As a sub-group, polls show that they are one of the most despised. In America, self-proclaimed atheists have practically no chance of election to high public office (there is one recent exception to this in the US Congress). Because of this, and because "atheism" is a negative descriptor (it defines what they are not, but says nothing about what they are) many atheists may prefer being called secular humanists, philosophical naturalists, philosophical materialists, non-religious, or skeptics.
Atheism can be the endpoint of a long transition from religious to deist to "spiritual but not religious", maybe a quick stop at pantheism, to agnostic, and finally to atheist. Many on the de-conversion journey are not comfortable making the leap in one fell swoop, but take time to transition. It can take a while to unlearn what accrued over a lifetime.
Up until about the time of the Enlightenment in Western Europe, a label of atheism didn't necessarily indicate lack of belief, but more a lack of willingness to accept God's rule. Descartes, who is famous for basing his philosophy on a proof of God's existence and benevolence, was accused of being an atheist by some of his contemporaries, mostly because he didn't believe in god the way you were supposed to. It was not until relatively modern times that Westerners actually started to commonly think about rejecting the actual existence of a god or gods. Prior to this, Protestants called Catholics atheists, and vice versa, and both called the Hindus, Moslems, and other "heathens" atheists. Theists considered deists to be atheists. It was thrown about much as the epithet, "communist", was used as a general purpose ad-hominem attack during the second half of the 20th century, and how "bigot" and "racist" is frequently used in the 21st century. It used to have more of a connotation of "incorrect belief" rather than "no belief", as it does now. There were early schools of atheist philosophy in ancient Hindu and Greek teachings, but only in the last couple of centuries has the movement gained momentum.
With the coming of the Enlightenment, or Age of Reason, began an era in Western civilization which emphasized logic, reason, and empiricism as the primary sources of legitimate knowledge. The enlightenment was a movement toward science, knowledge, and reason, and away from religion, ignorance, superstition, and dogmatic acceptance of "established" philosophical "facts" from the Greek philosophers (particularly Aristotle). The Enlightenment embodied a desire for human affairs to be guided by rationality and evidence rather than by faith, superstition, or revelation; a belief in the power of human reason to change society and liberate the individual from the restraints of custom or arbitrary authority; all backed up by a world view increasingly validated by science rather than by religion or tradition. The Scientific Revolution, which roughly coincided in time with the Enlightenment, amplified this trend.
Related to atheism is non-theism, which is the belief that the universe can be explained without any reference to the supernatural, or to a supernatural being. This is similar to metaphysical naturalism. Some non-theists avoid the concept of God, while accepting that it is significant to many; other non-theists understand God as a symbol of human values and aspirations. Many schools of Buddhism may be considered non-theistic.
Some atheists contend that there is no compelling evidence for god, and insufficient reason to believe. On top of that, there is insufficient "prior probability" to continue to suspend judgment on the issue (as agnostics do) due to its inherent implausibility. Their reasoning goes something like this: For all past generations no clear evidence has been presented (short of personal testimony, questionable documentation, and muddled reports of miracles). We have no reason to anticipate any new compelling evidence is forthcoming anytime soon. Therefore, we are justified in inferring that god probably doesn't exist. This line of thought brings in the concept of reliable knowledge vs. certain knowledge. Although we don't have absolute and certain knowledge that a god doesn't exist, we have reliable knowledge that this is the case. In other words, we can be as certain of this as we can of most things we encounter in life. Just as we cannot say with 100% certainty that the sun will rise tomorrow or that a real world exists outside our front doors, we can (with a high degree of confidence) make these assertions. The same can be said of a confident lack of belief in the existence of a god. To paraphrase from Stephen Jay Gould's description of scientific facts, atheists can't have "absolute certainty" of god's non-existence. They can only say that it is "confirmed to such a degree that it would be perverse to withhold provisional assent."
So, does "absence of evidence mean evidence of absence"? It can, if the evidence was strenuously and diligently sought after, but was not found. However, an omnipotent god is outside the bounds of experiment. He can always befuddle, outsmart, and outmaneuver any experiment designed to detect his presence. This type of "special pleading" (god is there, but you just can't see him) is very convenient for god's defenders, and is used in other contexts to support belief in ghosts, ESP, UFOs, hollow earth, and other fringe entities and belief systems.
Agnostics haven't gone as far as atheists in their declaration of the non-existence of gods. They claim that there is not enough evidence either way to make a decision. Agnostics are frequently criticized for being wishy-washy for refusing to take a stand. Some agnostics are really atheists who are not ready to "come out" in public. Some honestly think that there is not enough evidence to reach a conclusion one way or the other.
Ignostics (a lesser known term, synonymous with Theological Noncognitivism) object to the vocabulary, terminology, questions, and concepts used in religious discussions. They claim that there really is no coherent definition of God, and that there are too many contradictory and irreconcilable definitions to make sense of. They assert that, without a clear definition of what is meant by "god" in "does god exist?", we are putting the cart ahead of the horse and probably asking the wrong questions. Without knowing what we are asking, we will end up almost certainly just confusing ourselves with misleading terminology that seems to make sense, but really doesn't. As Wittgenstein recommended, they prefer to "pass over in silence" this topic because it really can't be discussed reasonably, and the propositions that are made about it one way or the other have no "truth value", neither being capable of proof nor disproof. In other words, it is a useless exercise to speculate on the ontological status of a "god". In this they echo Buddha when he outlined his "unanswerable questions" - a set of questions that can't be answered, may not even make sense to ask, that waste our effort, and whose pursuit misdirects our energy and focus from more important things.
None is a relatively new term that belongs in this chapter. It has been described, tongue-in-cheek, as the fastest growing "religion" in the West. It mostly shows up in younger people living in First World countries in the West. It describes people who, when answering a questionnaire on religious belief fill in the box marked "None" or "No religion" instead of Christian / Hindu / Moslem / Buddhist / etc. They don't identify with any of these groups, and would not affiliate themselves with these or any other groups that might show up on such a list (including Atheist and Agnostic). They resist being categorized and labelled in this manner, and see no reason to submit to the process. Because of this, it is difficult to make general statements about what they are "like" or what the believe or disbelieve. I think it is fair to say that they don't really want to participate in a discussion or debate about the topic of spirituality, salvation, praying, or any other religious topics. They don't want to be thought of in religious terms, and they don't use religion as a framework for how they approach life. They don't attend church very much, if at all. Thoughts about god just don't play a significant role in how they move through their lives. Many do confess to believing that a god or "higher power" may exist, and may even technically belong to a church (that is, they never officially separated from whatever church they were brought up in). Some admit that they occasionally pray. But for the "Nones" god and religion play only a small role. It it indicative of the lessening relevance of religion among many younger Europeans and Americans. Looking forward, the "Nones" are unlikely to join churches, buy bibles, give money to churches, have their children attend church, get their children baptized, or raise them in a particular religion.