First, I think we've all most commonly seen the "colloquial" form of Nihilism, which is a sort of despair over a lack of meaning in life. I'm pretty sure most people are aware that this is oversimplified, but it is partially grounded in actual Nihilism. The emphasis is put on despair, but the despair isn't the philosophy, just a product of it.
To sum up Nihilism in a sentence: reality lacks inherent meaning, and any meaning we attempt to fulfill is ultimately chosen, not given to us. Importantly, most of the sense of purpose we see as being "real" comes from our desire to continue existing, (hence Existentialism.) But, existence is a choice. You simply choose to continue to exist, or you don't. (In which case you die.) This is the main insight which makes Nihilism disturbing. Philosophically, if our existence has no purpose, then it seems that everything else we believe and value is built on a foundation of sand.