Hardcore dancing is performed at hardcore shows, to hardcore music. Or if so inclinded, just to hardcore music in your backyard.
It was first made in the early 80s when the genre of hardcore-punk developed from traditional punk. The hardcore punk scene started out with a lot of straigh-edge kids, who dedicated themselves to human rights. Those who who hardcore danced had no intention of injuring people or causing shit in the pit. They were merely expressing the deep respect they had for the band they were listening too, and showing their emotions through dancing. Most of these most involve what looks like flailing, it is more so a fighting. All of the moves have a meaning, and the fight moves represent fighting facists and those who supress the scene (such as people who have defined this word ignorantly, and negatively).
Others may perform moves to represent their humble attitude.
True hardcore kids can have a good time anytime, if a kid falls they will pick him up, if someone wants to start shit with them, they'll make a good effort to calm them down.
It is very unfortunate to see something degrade into such a steriotypically hated ignorance.
arm mills - swinging arms around, whether side to side, in circles, etc
picking up change - jumping back and forth with the intention of picking up the change of careless rich people
two-step - similar to skanking with skilled leg swinging motions.
spin kick - a simple "cresent" spining kick