To have hard fellings usually means to feel anger or resentment towards somebody. 'No hard feelings' is usually used to say you no longer feel anger or resentment for someone you may have argued with, fallen out with or who has upset you. 'So we are friends again, are we? No hard feelings?' You can also say 'No ill feelings'. No hard feelings can also be used to mean 'no problem, I'm not angry.'
'Bad blood' and 'ill will' also have a similar meaning although we don't use them in such an informal way.