It's characterized by naturalistic acting and dialogue (sometimes improvised), low-budget film production, an emphasis on dialogue.
John Cassavetes is the first director I thought about, especially "Minnie and Moskowitz" and probably all his 70s stuff, but also Vincent Gallo's "Buffalo '66", where you have Ben Gazzarra playing the father, who was part of the Cassavetes troupe, which is why I think Vincent Gallo got him on board.
funny ha ha is usually held up as the film that spawned the movement. i just rewatched it a few weeks back & was very pleased at how much i still really enjoyed it.
that, & the other mc films of that era - mutual appreciation, puffy chair, all that stuff - definitely took huge inspiration from cassavetes, no question. & it's not like those films all came out of nowhere in the 2000s. there was already a big subset of indie films in that realm. the things that you associate with mc -amateur actors, some improvisation, no lighting, naturalistic - could just as easily apply to something like clerks, for eg. or clean, shaven to give another eg.
mubi has quite a nice selection of those kinds of films, & i've had a good time revisiting them recently.
the color wheel
funny haha
mutual appreciation
the puffy chair
daddy longlegs
sun don't shine
all the light in the sky
listen up philip
the overnight
beeswax
dance party, usa
i don't mind it, in that it's a simple short-hand way to categorize a type of film that i generally really like. the same way i tend to enjoy bands that fall under the sadcore/slowcore banner. the names are silly, i guess, but as soon as i hear that something's been made that falls under that umbrella, i immediately know it's something i'm keen to see/hear.