Most of the time when someone is referring to "hardcore" they are referring to hardcore punk.
The word "hardcore" is defined by the American Heritage Dictionary as:
a form of exceptionally harsh punk rockHardcore punk is a subgenre that is heavier and faster than "traditional" punk rock. Songs in this genre are usually very short, very fast and very loud.
Some of the bands that are considered pioneers of this movement are Black Flag, which formed in Los Angeles in 1976, Bad Brains, which formed in Washington DC in 1977, and Minor Threat, which formed in 1980, also in Washington DC.
Other notable bands of the hardcore punk genre are The Angry Samoans, JFA, Agnostic Front and Sick of It All.
Some "essential" albums of this genre (in my opinion) are:
Black Flag's compilation album The First Four Years
Minor Threat's compilation Complete Discography
Bad Brains' 1982 self-titled album.
You can find out more about this genre at the Wiki article here: