I was just watching While You Were Sleeping and she says: 석 달전 쯤에 꾼 꿈인데..
I'm confused about what 석 is and when it's used as the number three. Are there equivalents for other numbers as well?
석 is one of the old quantifiers conventionally used by the elder people in Korea. There are many old quantifiers in Korean. For example, other than 세, there is also 서 which has the meaning of three. Some old Korean proverb:
구슬이 서 말이라도 꿰어야 보배
→ (literally) It takes more than three 말 (18 liters) of pearls to make a necklace.
→ Nothing is complete unless you put it in final shape.
The Rules of Standard Korean Language (표준어 규정) say that nouns ~돈, ~말, ~발, ~푼 must be used with 서 (such as 서 말 and 서 푼) and ~냥, ~되, ~섬, ~자 has to be used with 석 (such as 석 냥, 석 되). However these nouns are quite old words, and one might encounter them in some Korean historical movies or dramas.
Other old quantifiers include 너 (four), 닷 (five), 엿 (six) etc.
Nowadays, 세 is used more often and actually is a recommended form of the The National Institute of Korean Language (국립국어원).
'세', '석'이 단위 명사와 자연스럽게 어울려 쓰인다면, 둘 다 표준어로 인정하되, '세'를 원칙 표기로 보고 있습니다.
If both '세' and '석' is used naturally with nouns having a meaning of unit, both are standard Korean, but '세' is recommended.
So both 석 달 and 세 달 are standard Korean, but Koreans use 세 달 more often.
세 달 (석 달) and 삼 개월 are almost identical in meaning, just that words originated from Hanja (삼 개월: 三個月) have more formal nuance. In addition, maybe 석 달 has a feeling of a person above 40s is speaking (just my opinion).
셋 달 is awkward since both 셋 and 달 are nouns. In Korean, when saying about some quantity, we have to use words of 관형사 (similar to adjectives in English). For example,
세 사람, 네 그릇, 두 컵
Three people, four plates, two cups
하나, 둘, 셋, 넷 are used as nouns in sentences:
셋은 앞으로도 늘 함께 지내기로 맹세했다.
The three pledged to be always together from now on.