I mentioned before that I think Nihilism gets a bad rap. Now I'm going to talk about it.
Nihilism is generally associated with Friedrich Nietzsche and Heidegger. Oh, and Nazi Germany because of things like "The Will to Power" and _übermensch_ (superman). However, they're a lot of other things on top of nihilism.
At it's core, nihilism is belief in nothing. Moral nihilism is the belief that there exists no objective moral truth, no moral authority, and no morality. One typical theist assertion regarding atheism is that being an atheist necessarily makes you a nihilist, and this brings along with it the above-mentioned associations. It is followed by the assertion that "Since without God you can't have morality, then without God you can't be good".
Does atheism lead inevitably to nihilism? No. For example, Sam Harris (one of the Four Horsemen of New Atheism) argues that there is an absolute moral framework absent God based on decreasing suffering and increasing happiness, a form of utilitarianism.
However, many atheists are indeed nihilists, myself among them. I believe there is no absolute morality and no external moral authority. But that doesn't mean I'm either immoral or amoral. I do believe there is morality. The work of Jonathan Haidt indicates that human morality is universal and follows certain trends. Therefor, I posit that human morality arises from biology. More on that later. Nihilism and amorality aren't actually the same thing.
In other words, atheism doesn't lead to nihilism, and nihilism doesn't lead to villainy.
There's a reason the association of nihilism with Nietzsche is seen as a negative, particularly by Christians. Nietzsche was unabashed in his criticism of Christianity and his support of nihilism. Memorably he said "God is dead". Nietzsche also described two forms of nihilism. Passive nihilism, or the will to nothingness, as a sort of Western Buddhism; the abnegation of self. Active nihilism, which he advocated, seeks out and destroys structures such as Christianity. This wipes the slate clean and allows the construction of a new structure.
Finally, he described a typical response to discovering nihilism. On learning that a certain belief system is false, the newfound nihilist despairs and begins living his life in rejection of what he used to believe. For example, a follower of a faith that preaches abstinence from sex and drugs will begin boozing and screwing. Instead of preaching good, he may exult in villainy. And, as an added bonus, there's none so zealous as a convert. Many like to say that Nietzsche was advocating this sort of life, when in fact he thought that sort of person was weak.
Which brings us to my own personal take on different kinds of nihilists, weak vs. strong. Whereas for theists and atheists, the use of weak and strong weren't meant to imply a value judgment, this does. A weak nihilist is like the one described in the previous paragraph. He sees that there is nothing and lets it overwhelm him; the nothing overwhelms him. The strong nihilist is like Nietzsche's active nihilist, only instead of trying to destroy what came before, he's trying to build something new. He sees the nothing and seeks to fill it.
If you want another good example, see Angel's arc in the second season of his own show. First, he descends into darkness and eschews the company of his friends. He exults in violence and destruction. Yes, he kills evil, but he's not nice about it. This reaches its nadir when he learns that the villainous law firm doesn't take its orders from hell. "The world doesn't work in spite of us, but with us." Then he has an epiphany.
Angel: [It doesn't m]ean anything. In the greater scheme, in the big picture, nothing we do matters. There's no grand plan, no big win.
Kate: You seem kind of chipper about that.
Angel: Well ... I guess I kind of worked it out. If there's no great glorious end to all this, if ... nothing we do matters ... then all that matters is what we do. 'Cause that's all there is. What we do. Now. Today. I fought for so long for redemption, for a reward, finally, just to beat the other guy. But I never got it.
Kate: Now you do?
Angel: Not all of it. All I want to do is help. I want to help because I don't think people should suffer as they do, because if there's no bigger meaning, then the smallest act of kindness is the greatest thing in the world.