It means something that is cute in Korean. This word became popular over internet because of a Gwiyomi Song, which is a very cute and catchy song, featuring some mathematics. 1+1=gwiyomi, 2+2=gwiyomi etc.
This word is frequently used as an answer to a mathematical, chemical or other type of question and equation when you don't know the answer. Or if you just plainly dkn't want to answer something.
friend: how much is 23×65?
me: um... gwiyomi?
friend: you'll never pass this test.
me: hey, what's your weight?
friend: gwiyomi.
me: ok, I get it. I'm never gonna ask that again.