Authentication is the moment a system asks you "are you really you" and demands proof. Here's exactly what happens then.
Picture the entrance of a building with security. At the desk they ask for your ID card before handing you a pass for the elevator. The ID card proves who you are. The pass shows how far inside you're allowed to go. Authentication is exactly that check at the entrance - only instead of a desk it's a screen, and instead of a card it's a password or a code.
When you log into your mail or your online banking, you type in an email and a password. Those are your credentials - your ID card in front of the system. It checks them against what it has on record, and if they match, it lets you in. If someone else guesses your password, as far as the system is concerned, they are you. That's why a password has to guard something only you know.
A second factor is a second check at the same entrance. After the password comes a code sent to your phone, or a confirmation in an app. It's as if the guard, besides the ID card, also picks up the phone and calls to check it's really you. Even if someone has stolen your card, without that second call they don't get in.
The problem is rarely a broken lock on the front door. The problem is a side entrance nobody put a lock on - because nobody thought that was exactly where a person could get in. The system didn't ask for an ID card, because nobody decided it should ask there.
Here's what holds
Building systems myself, I see it from the inside. It's easier to put a sturdy door up front and forget the side entrance, because it looks unimportant. Every place where data comes in has to ask who you are. Not because we don't trust anyone, but because the system shouldn't have to guess.
And here's the honest answer for why everything revolves around authentication. Not because it's a trendy word. Because it's the only way a system can tell the difference between you and everyone else in the world.