Authentication is how an application proves who you are, and broken authentication is the family of flaws that lets an attacker defeat or bypass that proof. It covers a lot of ground: no rate limiting so passwords can be brute-forced, weak or default credentials, predictable session tokens, and flawed password-reset flows that let one account take over another.
The most approachable example is a login form with no protection against repeated guesses. If the application accepts unlimited attempts and users choose weak passwords, an automated tool can simply try common combinations until one works. This is where tools like Burp Intruder and Hydra come in, and it is why rate limiting and multi-factor authentication matter so much.