Grieving and mourning are both related to the death of a person or a pet.
Grieving is usually more serious and occurs closely after the death. "Grief counseling" is a therapy service for people coping with the death of a loved one. Grieving is normal, but grieving can be a stressful and traumatic experience. People going through grief may stop taking care of themselves, not sleep, not eat enough. They may require the care of a professional like a therapist.
Mourning is usually over a longer period of time after a death. Usually, a person in mourning is back in the routine of daily life, but they may be still in low spirits.
Regretting is sadness over something that you have done. It is a feeling of "I wish I hadn't done that" or "I wish I could go backwards in time and make a different decision."