Basically they're just very down-to-earth stories that are about young people going through the transition to adulthood.
Normally the story will be centred around a character who is still learning to be responsible and to think for themselves, and they will go through some trials that test their will and/or fortitude.
There's a lot of very loved films in the genre - films like Ghost World and Ladybird. They typically feature a character who is still figuring themselves out, and normally they have some kind of dream they want to follow but they're scared to.
Usually the coming-of-age story will follow them building up the courage to reach for this dream. In Ghost World, the protagonist's dream is to leave town and start a new life away from people who don't seem to understand her. In Ladybird, her dream is to move to a larger city to get a better education. Both characters are desperate for something that other characters in their story are obstructing them from. Both characters go through various trials (romantic trouble, loss of friendships, family strain, failure at something they're passionate about) and the story ends when they finally are mature enough to act on their dreams.
Normally the protagonists of these stories will go through tough character building experiences or even some mild trauma that they will have to process and heal from. Normally they have learned a lot and become a lot stronger by the end. Normally they have learned to be less afraid and hesitant, and have gained a sense of independence.
These stories are almost always plotted around and based on emotional journeys and mature subject matter and feature young and awkward protagonists. They're often set in very relatable situations and/or locations. They can vary a lot in tone, from more comedic to more serious, but these things are normally common between them.