The phrase 'The Dark Triad' refers to a situation where the same person has each of three 'dark' traits. These traits are: Narcissistic Personality Disorder, machiavellianism, and psychopathy.
Now, you already know about the Narcissistic Personality Disorder element of the dark triad. However, what about machiavellianism and psychopathy?
Machiavellianism means "the the employment of cunning and duplicity in statecraft or in general conduct." It is "characterized by a duplicitous interpersonal style, a cynical disregard for morality, and a focus on self-interest and personal gain". (Both quotes from Wikipedia.)
So, you can see from these descriptions that these traits and behaviours fit very well within our experience of Narcissistic Personality Disorder.
The third trait is Psychopathy. Now, the DSM V doesn't use this word, nor the word sociopathy. Instead, they label it Anti-Social Personality Disorder. But for our purposes, as we explore the dark triad, we can use the word 'psychopathy'.
Psychopathy, in the DSM IV at least is considered part of the same 'family' as Narcissistic Personality Disorder. They, along with Borderline Personality Disorder and Histrionic Personality Disorder are part of the Cluster B section. Psychopathy is therefore a 'cousin' disorder to narcissism, in a way.
And so, it's not surprising that psychopathy and narcissism often manifest together.
The Dark Triad
To my mind, The Dark Triad is saying the same things in different ways. We can call it Narcissistic Personality Disorder, or machiavelliasm, or psychpathy, or all three.
I, as an ordinary person, won't ever be able to officially diagnose anyone with The Dark Triad. Neither will I be able to diagnose them with any one of the three traits. And I imagine, neither will you (unless you are qualified of course!)
But in the end it doesn't matter. We don't have to decide if this person is specifically one or all of the elemenets of the dark triad. We just need to know that such people exist, and that they are very unpleasant, manipulative, remorseless, egotistical, empathy-less individuals.