Cannot: 못 before the verb

못 goes in the same place as 안: directly before the verb, with a space — 못 가요, 못 먹었어요, and in noun + 하다 verbs before 하다: 숙제 못 했어요. Said aloud it changes the sound that follows, because its final consonant meets the next word: 못 와요 sounds like [모돠요] and 못 먹어요 like [몬머거요]. The romanisation on this site shows each word on its own and does not show this.

안 says that something does not happen; 못 says that it cannot, whether for lack of ability or because circumstances get in the way. 오늘 파티에 안 가요 is "I'm not going to the party today" (my choice); 오늘 파티에 못 가요 is "I can't go to the party today" (I would, but something prevents it). Declining an invitation with 못 is the polite choice, because it shows that you would have liked to come.

못 only goes with verbs of doing, which somebody could be able or unable to do. It does not go with adjectives: a thing cannot be "unable to be expensive", so the weather is 안 좋아요, never with 못. For a fuller statement of ability there is 갈 수 없어요; 못 가요 means the same and is shorter and more conversational.

In the past, 못 means "did not manage to": 어젯밤에 잠을 잘 못 잤어요, "I didn't sleep well last night"; 시간이 없어서 아직 못 봤어요, "I haven't had time to see it yet". English speakers tend to say 안 잤어요 here, which sounds as if you stayed up on purpose. Note also the spelling: 못하다 written as one word is a separate verb, "to be bad at", treated in the section below.