Nu-Metal : A term given by the underground and adapted by the media, in regards to the "modern era sounding metal" bands such as: Korn, Disturbed, Limp Bizkit, POD, Linkin Park. This genre often contains hip hop or rap elements, as well as the turn table...also often found in hip hop or other afro-american cultural music. This is usually based on the common pop formula idiom and meant for radio air play and mass appeal.
Hardcore: A Genre of extreme (and typically juvenile) music based on hardcore punk and adapting elements of metal into it. The lyrics usually contain messages of self discipline and/or esteem, anti drug use/politics/relationship issues and usually in a violent or aggressive way. The music usually always features "breakdowns" which is a simplified and exagerrated portion of the song typically meant for the audience to dance in a very silly and violent way (typically an alternative to the "mosh pit".) Some example bands are: Hatebreed, Strife, Eighteen Visions, Throwdown, Zao and Earth Crisis. Usually fags listen to this genre.
Metalcore: A relatively new sub genre of metal which takes the recent influence of the "gothenburg metal" scene and melds it with the above hardcore scene in the sense that they take verses and chrouses based on melodic "at the gates or in flames" styled riffs then put the before mentioned "breakdowns" in it as well. This started in America, and continues to prominantly stay there...thank god. This genre, more often thought of as metal than hardcore, is liked by both subcultures, and seems to be the melting pot of them. Some example bands are: Killswitch Engage, Shadows Fall, Avenged Sevenfold, God Forbid, etc This genre is also terrible, please steer clear at all costs.
Emo: dont listen to this.
Ever.