싫다 expresses the negative emotion or reaction of disliking or being repelled by something. It means you just don't want to get involved with the person or thing.
- 밉다 describes a different kind of dislike. You dislike someone actively with your mind occupied by it, like a resentment or grudge against someone (it is not used for inanimate things). We commonly use it for a mild (sometimes even cute), passing emotions of such kind.
These words are adjectives and have a special usage for the first person in the present tense. For example, "I dislike dogs"
is said as 나는 개가 싫다 with no verb but only adjective. It's a little like 나는(for me) 개가(dogs are) 싫다(loathsome).
This is a very direct and vivid construct reserved for the first person.
For other cases, we use the corresponding verbs with the same meaning, 싫어하다 and 미워하다, as in 동생은 개를 싫어한다.
This adjective vs verb distinction applies to a set of emotion words, like 좋다, 싫다, 기쁘다, 부끄럽다, 아쉽다, etc. It can be difficult to grasp in the beginning.
- 나는 귀찮다는 듯이 대하는 남편이 미웠다 - I resented my husband who acted as if he couldn't be bothered.