面子 ('face' = social standing, reputation) is a central organizing concept of Chinese social life — sociologist
林语堂 (Lin Yutang) called it 'the most delicate standard by which Chinese social intercourse is regulated'. Losing face (
丢面子) means publicly forfeiting respect; giving face (
给面子) is letting someone save it. Compare
没面子 (not have face),
要面子 (need face / be vain),
撑面子 (keep up appearances).