항아리 is a kind of "jar," and 손님 means "guest," so together they mean "jar guest," which is one name for the infectious disease, "mumps." Koreans also refer to the mumps as 볼거리, and Korean doctors refer to it as 유행성 이하선염. Supposedly, the word 항아리 (jar) is used in the name because mumps causes one's cheeks to swell up like a 항아리, which is a jar that has a mouth smaller than its belly.
손님 (guest) is also used in the names of other infectious diseases. Small pox is referred to not only as 천연두, but also as 마마, 손님, 손님마마 and 큰손님 (big visitor). Chicken pox (수두) and measles (홍역) are also referred to as 작은마마 and 작은손님 (small visitor). The names show respect for the diseases, as if they were honored guests.
Koreans used to believe the diseases were actually gods or spirts who were offended with them, so it seems the honored names were an attempt to appease the spirits or to avoid offending them in the first place. Since small pox was referred to as the "big visitor" (큰손님), it seems it was feared more than the other diseases.