noun
A somewhat formal or literary noun for deep regret or remorse over one's own actions. Often used in set phrases like 悔いが残る or 悔いのない人生.
あのとき正直に話さなかったことが、今でも悔いとして残っている。
I still regret not having spoken honestly back then.
悔いのないように、全力を尽くします
I will give it my all so that I have no regrets.
後悔 is the common everyday word for regret, used as both a noun and a する-verb. 悔い is more literary and often appears in fixed expressions.
Derived from the verb 悔いる (くいる, 'to regret'). The noun form 悔い is the continuative stem (連用形) of the verb.