계란지단 is a garnish (고명) made from chicken eggs that is used on a lot of Korean dishes, especially on Korean noodle and soup dishes. There is a white garnish made from the white of the egg and a yellow garnish made from the yolk.
To make the garnish, one must first separate the white and yoke of the egg and then fry them separately into paper-thin patties which are then peeled from the frying pan and cut into spaghetti-thin strips. The video below shows how it is done.
As for the name 계란지단, the Sino-Korean word 계란 (鷄卵) literally means "chicken (鷄) egg (卵)," and so does 지단 (鷄蛋), but "지단" is the Chinese pronunciation of "chicken (鷄) egg (蛋)"; the Korean pronunciation would be 계단 (鷄蛋). The Chinese character 鷄 (계) means "chicken," and both 卵 (란) and 蛋 (단) mean "egg."
So, the Korean word for "egg garnish" literally means "chicken egg, chicken egg."