Somebody gives you a baseball signed by Nolan Ryan. Is it authentic? You meet someone at a party, and you get the feeling that the person doesn’t seem very authentic. You enter your username and password into a website, and thereby become authenticated.
The dictionary definition of ‘authentic’ is “of undisputed origin; genuine”.
So, authenticity has to do with origin stories. If the baseball’s origin story involves an event where Nolan Ryan signed it, it’s authentic. If its origin story involves someone else signing Nolan Ryan’s name, it’s not.
Likewise with people: if the words and behaviors a person is exhibiting have their origin in that person’s real beliefs and emotions, we say the person is authentic. If there’s a layer of dissimulation in the middle, a sense in which the person is trying to appear to be a certain way, without actually being that way, then we say that the person is inauthentic.
To be authentic as a person implies that your thoughts and actions are genuine — that they are illustrative of the the ‘real you’. This doesn’t mean that you speak every thought that comes to your mind. It just means that when you do speak, your words are revealing a genuine part of what it’s like to be you.