If you're a foreigner living in Korea and want to invite your family, here are the requirements that you need to get a visa.
1. The person that you are inviting need a passport, the copy of his/her passport (front page) and a passport picture.
2. A copy of your passport (front page). You and your husband if you're married to Korean.
3. An invitation letter written in an A4 paper. You don't need special document or paper for that. Just write on a piece of paper the reason why you're inviting him/her and then your sign at the bottom page.
4. A copy of your NSO marriage contract if you're a foreigner married to Korean.
5. A marriage certificate from the government office in Korea.
Send the copies of #2,3,4, and 5 to your family to present at the Korean embassy including your family's requirement (#1). Goodluck!