One primary understanding of nihilism which existed even within early Russian Nihilism and has continued down through today is the idea that moral norms, but especially traditional morality, cannot be justified by any rational or scientific standards. As a consequence, they have no "reality" — they do not exist anywhere except in the minds of people and hence do not really "exist" at all.
In this sense of nihilism, it is arguable that nihilistic beliefs are very common today, especially among philosophers. A great many people regard moral statement as arbitrary, by which I mean that they either do not stem from some eternal and infallible source (like God) or they cannot be perfectly and absolutely grounded in unquestionable logic and reason. Indeed, there are so many positions which can justifiably be called moral or ethical nihilism (although ethical relativism is in more common usage) that it wouldn't be possible to describe them all here.
Instead, it would be better to simply take a look at the two primary positions which reject moral nihilism. The first and historically most common is the one which argues that moral standards are derived from God. God says what is right and what is wrong, God informs us about this, and we are obligated to obey what God has decided.
People who adopt this sort of position consider moral nihilism to be truly anathema — for them, the nihilistic position would license any sort of moral outrage and every sort of criminal behavior. This particular condemnation has been perhaps best expressed by the character of Ivan in Dostoyevky's Brothers Karamazov, who said "If God does not exist, everything is permitted" — an attitude which led him to acquiesce to his own father's murder.
This idea that morality requires divine sanction and support is simply not tenable anymore. The position is either false because this god is relying upon an independent standard of goodness, or it meaningless and circular, because this god's decisions about what is and is not moral is no less arbitrary than our own and what is "morally good" loses any independent definition.
It is largely because of this, I think, that modern critics of ethical nihilism rely less upon appeals to God and more on appeals to reason. According to them, justifiable moral standards must be based upon universal standards of reason and logic. How, then, should we resolve moral dilemmas where two sides conflict? It is entirely possible for two groups of people to sincerely hold incompatible moral positions, yet there seems little recourse to a "universal standard" that all should be able to accept.
It would perhaps be preferable if there existed a set of universal and objective moral standards that were independent of our emotions, prejudices, and passions — but even so, who knows if we humans would be able to find, identify, and understand them? It seems much more likely that ethical nihilism is the pragmatic position to adopt. If universal moral standards exist, we may have just as much luck hitting upon them under ethical nihilism as we do under moral realism.
