A pairing between two characters from the anime Axis Powers Hetalia of Japan and England. AsaKiku comes from the Japanese pronunciation of England's human name, Arthur Kirkland, which is pronounced with a Japanese accent as "Asa" (A-tha, silent r) and Kiku, Japan's human name (Kiku Honda).
Other "rival" pairings might include but are not limited to:
USUK, Giripan, FrUK, Ameripan, etc.
Person 1: I love AsaKiku <333
Person 2: Meh, I think that USUK is way cuter.
-cue shipping wars-