추다 means the verb 'to dance' and 춤 is the noun 'a dance'. 춤 is actually the noun form of 추다, as it is formed by adding the ending (으)ㅁ to the verb stem. I was told that we should say 춤 추다 'to dance a dance' and that 추다 alone is incorrect.
Hence my question: why?
In a similar way, 'to dream' is 꿈 꾸다, which is literally 'to dream a dream'. You can also read sometimes 잠 자다, 'to sleep a sleep', but it doesn't seem to be mandatory ('Good night' is 잘 자요, not 잠 잘 자요).
I don't know about the "why" part, but one way to think of them is that these verbs require objects. And it doesn't have to be the exact word 춤 or 꿈. For example:
축제에서 디스코를 췄어요. [I] danced a disco(?) in a festival.
사람들은 음악에 맞춰 왈츠를 추었다. People waltzed to the music.
지아는 악몽을 꾸었습니다. Jia had a nightmare.
...although I think "무용을 췄어요" is not used: we'd say "무용을 했어요".
I think we can draw a parallel with phases such as "전화를 걸다": here, the object is also mandatory. A phrase like "친구에게 걸었어요" would be usually not allowed, unless it's an answer to a question like "누구한테 전화를 걸었니?". Similarly, "어젯밤에 꿨어요." is normally not allowed, but it's OK if you were asked "그 꿈을 꾼 게 언젠가요?".
Just like any other transitive verbs, the object slot becomes "empty" if the object is used to build a relative clause:
그 사람이 춘 춤은 참 인상적이었다. His dance (= the dance he did) was very impressive.
내가 어젯밤 꾼 꿈은 참 이상했어. The dream I had last night was very strange.
Adding to @jick's great answer, there is a similar construction in English where an object of a transitive verb is a noun form of the verb. For example:
Live a happy life: 행복한 삶을 살다.
Dream a bad dream: 안좋은 (나쁜) 꿈을 꾸다.
Dance a wild dance: 격렬한 (광란의) 춤을 추다.
Breathe a last breath: 마지막 숨을 쉬다.
As you can see, '삶', '꿈', '춤', '숨' are noun forms of each verb. It is not a repetition and they are needed as the verbs require their objects.
You need to note that '꿈꾸다' and '잠자다' both are one word, hence there is no space in the word. You could write '숨쉬다' but, '숨(을) 쉬다' seems to be the correct phrase according to Naver Korean dictionary. '삶살다' sounds very weird as it is not a word.
