Being "remoseful" means feeling sorry for something that you did.
The word "remorseful" is more formal than "sorry". You can use it when you really want to let listeners know that you feel bad about something that you did, in a formal situation. For example, a lawyer might say this in a courtroom during a trial:
My client is truly remorseful for his actions.
The noun version of "remorseful" is "remorse":
Do you have any feelings of regret or remorse about what happened?