Something that is "hard core" is really strong and concentrated. For example:
I did this really hard core aerobics class last night, and today I'm just dying!
Don't you think parents should have the right to protect their children from violent images and hard core pornography?
A "hard core fan" is someone who supports something really actively and with a lot of enthusiasm. If you never miss a game for your favorite sports team and have the names of all the players memorized, you're a hard core fan.