Refers to a loss of reputation or a stain on one's honor. Often used in phrases like 名折れになる (to be disgraced) or 名折れだ (it's a disgrace).
If you do such a thing, it will bring disgrace on the family.
His misconduct is a dishonor to the team.
Compound of 名 (na, 'name, reputation') and 折れ (ore, 'breaking, loss'), from the verb 折れる (oreru, 'to break'). Literally 'a breaking of one's name', referring to a loss of reputation.