Deign is pronounced like 'Dane' - it rhymes with 'rain', 'train' and 'plane', IPA: /deɪn/. It means 'to condescend', 'to allow oneself graciously', or 'to lower oneself' to do something like accepting a gift, or to give an explanation. (It is the opposite of 'disdain', with which it is etymologically connected. 'Deign' comes from the Latin dignāri, 'to consider worthy', and 'disdain' from dēdignāri, 'to consider unworthy, to scorn or reject', both verbs formed from the adjective dignus, 'worthy'.)
See also -eign.