I want to hurry up and finish the intro/expository posts and talk about real material. So this is the last "intro" post on this topic, and in it I will describe several groups that overlap with atheists. Note that you need not be an atheist for any of these labels to apply. Anyone who hangs in the skeptical blogosphere probably already knows this stuff, but not all my readers do.
This is, of course, my personal favorite. I've already described it here. It's similar to rationalism, with an emphasis on science. To the skeptical atheist, religion is just one more claim without evidence.
Secularism actually does not mean atheism or theism, but neutrality between all religions. If I call myself a secularist, that usually means I advocate separating church and state. Everyone, including the religious, is better off with a secular government.
Humanism is an ethical philosophy based on the dignity of all humans. Most Humanists I see are part of a subcategory called Secular Humanism. Secular Humanism advertises itself these days as being the "positive" part of the atheist worldview. As the story goes, atheism is simply a rejection of gods, but you can't stop there because that is only a negative claim. Secular Humanists often recognize the good things that come out of religion (ie community, ethics), and try to provide secular alternative routes to these goods.
This refers to a specific group that follows the philosophy of Ayn Rand. I can't say I understand it well enough to explain it, but I guess atheism happens to be one of their beliefs. Objectivists are always politically libertarian, which means liberal on social issues and conservative on economic issues.
This is supposed to be a very inclusive label that applies to anyone who doesn't answer the question "Do you believe in God" with a yes. Some people think nontheism and atheism are in fact the same thing, but I won't go into the semantics.
A freethinker is anyone who has rejected organized religion. Atheism is only a subset of this larger group, but I think it's a majority, or quickly becoming one. I described some other groups within freethought just the other day.
I already described this just the other day, but I'll just note here that agnosticism overlaps with pretty much every freethinking group.
Bright (no -ism form, as far as I know):
What a lot of words and labels! Doesn't it seem like atheists must like making new ones all the time? Well, in this case, they really did make up the label, in just 2003. The idea is to replace "atheist", with all its negative connotations, with a happy word, just like the word "gay" for homosexuals. The official webpage defines a Bright specifically as a naturalist, meaning no supernatural beliefs. I don't particularly like the word because, apart from being very contrived, I think it connotes arrogance, as if I thought I was brighter than everyone else. Regardless, some people disagree, and I see it used on occasion.
Now that you've seen all the common labels, you may mix and match them to apply to yourself. I picked atheist, agnostic, skeptic, secularist, nontheist and freethinker. Hooray for identity politics?