Mumblecore: An early-21st-century independent film movement. Mumblecore films are populated by characters in their late teens and early twenties who have difficulty expressing their confusion about love, work, and life in general (hence "mumble"; the -core suffix is most likely a nod to hard-core and soft-core). Practitioners of mumblecore (also spelled with a capital m) are sometimes called the mumblecorps.
In "A Generation Finds Its Mumble," New York Times arts writer Dennis Lim identifies Andrew Bujalski's Funny Ha-Ha (2002) as "the film that kicked off the mumblecore wave." Other films in the genre include Aaron Katz's Dance Party USA (2006) and Quiet City (2007) and Jay and Mark Duplass's The Puffy Chair. Lim writes that "mumblecore" was coined during the 2005 South by Southwest Film Festival by Bujalski's sound mixer, Eric Masunaga.
For more on mumblecore, see this article in IndieFilmPedia.
Update: Erin discusses -core as a snowclone morpheme on her blog, Snowclones.org.