What sort of disease is a weak or short cervix?

Depends. Sometimes the cervix is just that way naturally. Sometimes it has been changed by surgical rpocedures. A short or weakened cervix is concerning for retaining a pregnancy.
Incompetent Cervix. A short or incompetent cervix is only a problem in pregnancy. See your doctor early if you've had pregnancy losses (after 12-14 wks) without contractions, had surgery on your cervix, or a short cervix on ultrasound. Management may include surgery (cervical cerclage), rest or both. Your OB can help you understand what's best for you.