Here is how I define atheism: atheism is the belief that there is no god. Now for the fine print:
1. The most common misperception people have about atheism is that atheism is certainty that there is no god. Although those who profess certainty that there is no god do indeed qualify as atheists, they are only a subset of atheists. Most atheists believe there is no god, but do not profess certainty about the matter. How can one be an atheist and still admit the possibility of being mistaken? In exactly the same way that a person can be a believer and still harbor doubts.
2. There is a special kind of person, called a noncognitivist, who technically does not believe that there is no god, yet still qualifies as an atheist. Noncognitivists argue that the word god is meaningless, such that to profess belief that there is, or is not, a god is to utter literal nonsense, on par with saying that you believe that there is, or is not, a blawdjaskid. Noncognitivists are relatively few and far between these days, so I think it best simply to understand them as a qualification to the definition I offered.
3. Among those who defend atheism by name, there are many who argue that atheism should be understood not as the belief that there is no god, but as lack of belief that there is a god. I think the real motivation here is to try explicitly to incorporate noncognitivists under the name of atheism. There are two problems with this: (1) such a definition ends up incorporating agnostics (those who just have no idea whether or not there is a god), most of whom do not consider themselves atheists; (2) very few people who call themselves atheists have a simple lack of belief that there is a god—on the contrary, virtually all believe that there are good reasons to actively deny that there is a god. Given this, I think it better to define atheism narrowly and then qualify it with noncognitivism, than to define it so broadly that it encompasses agnostics against their will.
4. Some people try to split the difference by calling those who believe that there is no god strong or positive atheists, and those who merely lack a belief in god weak or negative atheists. I think this distinction creates more problems than it solves. I prefer to reserve the title of atheist for those who believe that there is no god (and noncognitivists), and use the more generic term nonbeliever as the umbrella term for atheists and agnostics combined.
5. Since atheism simply is a stance on whether or not there is a god, it does not require any particular ethical or political views, or any broader beliefs about what does and does not exist. Atheism does not even require rejection of the transcendent or the supernatural in general. Atheism is not a worldview, but merely one element that may fit into many different worldviews.
6. With this said, modern atheists—at least those who openly identify themselves as such—tend also to be naturalists and skeptics, rejecting entirely the supernatural and even the paranormal.
7. Those who wish to define atheism differently than I have are, of course, free to do so. I think my definition best captures actual usage, but nothing critical turns on purely semantic debates; all that matters in a discussion is that everyone understand how everyone else uses their terms, so misunderstanding can be avoided. All I claim about my definition is that it captures what is atheistic about people who actually call themselves atheists.