안 (an) means you are not doing it. 못 (mot) means you are unable to. 안 가요 is I’m not going. 못 가요 is I can’t go, and in a lot of situations that gap is the difference between a refusal and an apology.
Both go directly in front of the verb, with a space. 안 먹어요, 못 먹어요. There is a longer form for each, covered below.
못 has a pronunciation problem worth learning early. Written mot, and the final consonant changes depending on what follows. Before a vowel it links: 못 와요 comes out mo-dwayo. Before ㄴ or ㅁ it becomes an n sound: 못 만나요 is mon-mannayo, 못 나가요 is mon-nagayo. Searching mot manayo will fail you.
The basic split
안 먹어요 (an meogeoyo) = I don’t eat that
Choice. Maybe you dislike it.
못 먹어요 (mot meogeoyo) = I can’t eat that
Inability. Allergy, or you physically cannot handle spice.
술 안 마셔요 = I don’t drink
술 못 마셔요 = I can’t drink
This pair is the one that comes up constantly at Korean dinners, and the two sentences produce completely different responses. 못 마셔요 ends the conversation, because nobody argues with a limitation. 안 마셔요 is a choice, and choices in a group setting can invite persuasion.
못 is the polite one
This is the part textbooks skip, and it matters more than the grammar.
Korean declining tends to reach for inability rather than unwillingness, because unwillingness places the refusal on you and puts the other person in the position of having been turned down.
내일 못 가요 = I can’t come tomorrow
Standard, unremarkable, requires no explanation.
내일 안 가요 = I’m not coming tomorrow
Blunt. Grammatically fine, and it will be heard as a statement about your priorities.
So when declining an invitation, 못 is the default even when the truth is closer to 안. Koreans do this constantly and it is not considered dishonest, in the same way that “I can’t make it” in English does not commit you to an alibi.
The long forms and where 않 comes in
Each negation has a longer version that attaches to the verb stem.
안 먹어요 becomes 먹지 않아요 (meokji anayo).
못 먹어요 becomes 먹지 못해요 (meokji motaeyo).
The long forms are more formal and more common in writing. In speech the short forms dominate.
This is where 않 (an) enters, and it is the source of a spelling error even Korean speakers make. 안 is a standalone adverb with a space after it. 않 is part of the verb 않다 and never stands alone. 안 해요 and 하지 않아요 are both correct. 않해요 is not a word.
Spacing matters too. 안 하다 with a space is the negation. 안하다 closed up is a typo, though you will see it constantly online.
Verbs that break the pattern
Some verbs cannot take 안 in front. 하다 compounds split instead: 공부 안 해요, not 안 공부해요. The 안 goes between the noun and 하다.
알다 (to know) is irregular in a different way. 안 알아요 is not said. The negative is 몰라요 (mollayo), a separate word entirely. 못 알아요 is also wrong, and the closest equivalent is 모르겠어요.
있다 (to exist) has 없다 (to not exist) as its own word, so 안 있어요 does not occur.
These three catch nearly every learner, and they catch them for years, because the general rule is so clean that the exceptions do not feel like exceptions.
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Both are neutral and belong at every level of formality. The long forms 하지 않습니다 and 하지 못합니다 are the written and formal-speech versions, and you will see them in reports and announcements.
The thing to be careful with is not grammar but the choice itself. Using 안 to decline something from a senior colleague is technically correct and socially heavy, since it names your unwillingness out loud. 못 with a brief reason attached is the standard shape, and the reason does not need to be detailed.
One thing I have never resolved: whether Korean speakers experience 못 in a refusal as a soft fiction or as literally true in the moment. When I say 못 가요 about something I simply do not want to attend, I do not feel like I am lying, and I could not tell you why. English “I can’t make it” seems to work the same way, so the phenomenon may not be Korean at all.