The usual definition is that you're being authentic when you are acting in accordance to your inner, desires, talents and needs, rather than those imposed on you by parents, peers, schools and work. There are all sorts of problems with that definition, starting with how you identify your inner needs, moving on to what happens if meeting those needs involves illegality or gross moral turpitude, and ending somewhere around what you do if you can't make money following your bliss.
It's a tricky idea, and there's no doubt that the management guru who swiped it was indulging in some semantic inflation when he used it. "Authenticity" is a technical term of existentialist philosophy. What the management guru means is "sincerity", "honesty", "independence of mind", and "lack of hypocrisy". Doesn't sound as sexy though. So what is it?
Well, as a start and to borrow an idea from J L Austin, "authenticity" is not a trouser-word: the concept of authenticity doesn't do the work, which is done by the associated negative concepts: fake, false, dissembling, put-on, made-up, forgery, going-through-the-motions and a hundred others. When we say something is authentic, we mean it's not any of the many varieties of fake. Attempts to describe authenticity in positive terms tends to vanish into hand-waving or tautology (such as "being authentic is being real"). So in a way, I rightly had no idea what they were all talking about, because the word has no positive meaning of its own. And a good honest plain English answer that is too.
Then I read a passage in Karl Jaspers' Philosophy of Existence: "what I myself am...always remains a question... My authentic self can never become my possession; it remains my potentiality. If I knew it, I would no longer be it, since I become inwardly aware of myself...only as a task." The last bit is easy: you are not a to-do list, nor the project manager of your life. You are more than that, even if you can't say what more. It was the first bit that was really interesting.
Suppose authenticity consisted in doing U, V, W, X, Y and Z. Suppose that those are well-described actions or dispositions (rather than vague, waffly hand-wavy gestures in words), so that you could learn how to do them. Then you can learn to do them whether you mean it or not. In other words, now you can fake authenticity. Errr, oops. Contradiction. So there must be something about authenticity that can't be well-described, some extra ingredient that can't be said, but only sensed or perhaps judged. Perhaps it's an attitude, a state of mind and a disposition of behaviour. None of which are things you can learn as a trick.
Authenticity isn't about some struggle between you and the rest of the world for the righteousness of your soul. I don't see why you can't be a graduate of Eton-Oxford-McKinsey and authentic at the same time. Nor do I see that being an unemployed dropout from some dysfunctional East London secondary school makes you any automatically authentic. It should be as possible for an accountant to be authentic as for a conceptual artist. It should also be possible to be an authentic asshole as well as an authentic Good Guy.
You are being authentic when you think, feel and act as a subject rather treat yourself as an object. Roughly. Acting like a subject means that your actions and reactions are based on what you need, want and desire, and how you think, feel and intuit at the time. Treating yourself as an object means that you deliberately check and filter your immediate reactions and urges and replace them with ones that will meet the approval of others, or will project or maintain an image of yourself you wish to project. Because you are not a psychopath or a borderline, one of the things you want is that you should not hurt or upset others needlessly: manners and tact are not in-authentic. This is the balancing act of authenticity: too little guard on your responses and you're being tactless and gauche; too much tact and self-management and you're being misleading, opaque and very possibly insincere. Similarly, too little planning and you're living in directionless chaos, too much and you're being your own to-do list. Authenticity is the constant struggle to resist falling into conventional response, managed behaviour and a daily routine on one side, and the chaos of immediate impulse on the other.
Authenticity requires self-knowledge and honesty. This makes it difficult for people who have been raised in dysfunctional families to be authentic, as they simply don't know enough about themselves. It is also difficult for people who have been brought up with grade inflation and politically-correct praise to be authentic, as they have an utterly unrealistic idea of their skills and knowledge.
A wife who feels attracted to a man she meets but does not sleep with him is not being in-authentic: she is keeping her marriage vows. If she denies to herself she wanted to shag him senseless, she's in denial, and denial is never authentic; but if she denies it to her husband, she's being tactful. Good corporate drones live in a state of perpetual bad faith, and the guy in the video store working on his film script crosses the line between hope (authentic) and delusion (inauthentic) hourly. Anyone who takes a job where they have to apply rules and can say "it's not me, it's the job" is as inauthentic as a seven-pound note. And Clairol Balsam Golden Blonde always crosses the line. After all, if you're going to lie about your hair colour, what else are you going to lie about?