A "serious" cut or wound is one that might cause a lot of health problems if it's not cured quickly. When you say that a wound is "serious", it means that it might need to be treated by a doctor.
"Serious" is used in a similar way to mean "quite bad" in some other phrases:
He's been in a serious car accident.
This is a serious problem.
Nuclear weapons are still a serious threat to our national security.
If you don't turn this in on time, I promise you that there will be serious consequences.