밉다 and 싫다 mean "dislike" or "detest" (or the colloquial "hate"), but there is a difference.
(증오하다 is the word for hating someone seriously)
- 밉다 - 1) happen to resent someone for a specific reason, 2) something looks bad or annoying (눈에 거슬리다).
- 싫다 - dislike someone/something, as a natural reaction or for a reason; be repulsed by them.
싫다 usually means simply not liking someone, with no further interest in them (나는 저 사람이 싫다).
밉다 can be simply disliking someone like 싫다. It can also mean harboring a transient resentment for someone (because of their behavior) you're interested in. Unlike 싫다, 밉다 can often be used about family members or friends or love interests, with a wish to see them changed to your satisfaction. It's a mixed, love-hate feeling in such cases.
e.g. 그녀는 그의 무신경한 태도에 미운 마음이 들었다 (She felt a resentment arising at his indifferent attitude).