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Atheism is the lack of a belief in a god or gods. We'll focus on monotheism here, since it's more prevelant.
We do not claim to know for certain that god does not exist, but are simply unconvinced that he does. We view god as an unfalsifiable hypothesis, much like elves or goblins.
Since god is not immediately apparent to any of our senses, evidence of him is needed if we are to give the idea of his existence any credence at all.
The primary assertion of atheism is not neccessarily that "there is no god," but rather, "there is no good evidence for a god." And until evidence of a god is presented, we will be atheists.
The beauty of nature is not evidence of god, unless the ugly aspects of nature such as disease, famine, rape and death are evidence against god. Atheists view nature as lacking conscious planning, and thus see both it's beauty and it's ugliness as incidental.
The bible, the koran, someone's personal sense of god--these things are not evidence, but articles of faith. Faith is the opposite of evidence. In fact, the very definition of faith is "belief without evidence"--which is what many atheists are really against.
But even these non-evidence-based beliefs wouldn't be a problem if they didn't influence societal opinions and attitudes and negatively affect public policy and our political discourse. With that said, it should be noted that not all atheists have qualms with other people's religiosity, but many do take issue with it for the very reasons I detailed moments ago.