The Manic Pixie Dream Girl is a character type trope that's a staple of saccharine sub-par romantic comedies and harem anime, though it pops up all over the place in various forms.
Tell me if you've heard it before:
There's this guy. Maybe he's an introvert, a homebody who doesn't like going out. Maybe he's unlucky in life and love both. Maybe he's broody. Maybe he's reserved. Maybe he's a workaholic who can't stand being away from the office. Maybe he's a widower who never got over losing his first love. It all boils down to the same thing: the dude isn't really living. At least, not in a way the target audience would find fulfilling.
But then he runs into someone. Maybe they meet cute and literally bump into each other. Maybe he's decided to change his life and they run across each other at whatever dance club or pottery class he's decided to attend. Maybe he gets robbed. Maybe they're at school and she's the new foreign exchange student.
Whatever the case, the male protagonist will come across a female character who is quirky, but in a way that's endearing rather than alienating. She's happy and vibrant and creative, and will usually have a childish naivete about the world. In her purest form, she sees things through rose colored glasses, and it seems that within the first ten minutes of meeting the protagonist, she will dedicate her life to getting him to see the world that way too, dragging him along on her adventures and breaking him out of his rut. There might be some hint of her own story, but if there is, it's glossed over or unimportant to the main plot. The real issue that the writers want to address is thus: boring guy needs a peppy emotional cheerleader girlfriend to help him appreciate life and its wonders.
There is also the Magical Girlfriend variant where the girl in question is actually an angel or robot or genie or mermaid or some sort of supernatural/nonhuman entity.
Depending on the level of cynicism on the part of the writers, the manic pixie dream trope can be deconstructed to show how that level of immaturity and impulsiveness is actually kinda bad in real life. That's where you start getting more horrific examples that involve men getting stalked by emotionally unstable women, or finding out that the chirpy, quirky girl is actually dealing with mental instability/depression and needs to be rescued after all.
In more modern media, there is also the trend of genderflipping the roles and having an adventurous and childish guy character make a woman appreciate life and all its mysteries and wonder. This is related to, but not the same as, the other overused bad rom-com trope of a tough workaholic lady who says she wants to be independent, but actually needs a charming man to come into her life and teach her love.
Wikipedia has an entire list of proposed MPDGs