The word Gwiyomi (귀요미) does not only mean "cute" as many translations predicate. "Cute" is only the translation of the last syllable mi (미). The first part Gwi (귀) can be translated as "ear" or "spoon". Combined with the second syllable Gwiyo (귀요) it is a diminutive form of ear or spoon which means little ear or little spoon. So the literal translation of Gwiyomi is "cute little ear" or "cute little spoon". The whole word Gwiyomi is an allusion on the German word "Häschen". A Häschen is a designation and describes a little cute young bunny. The allusion can only be understood, if you know, that in german language the ears of a bunny or rabbit are called spoons.
Look up 귀엽다 in the dictionary.
귀염 is a noun-form of 귀엽다, much like gerund in English.
-이 is an suffix which makes a word a person who is/do the word.
So, 귀염이 is a person who is 귀엽다 or who is a 귀여운 person.
귀염 and 귀염이 itself is a childlike and cute term that is usually used for youngsters.
귀요미 makes even cuter because of its pronunciation 요 than 여. Informal but widely spoken these days. There's a popular song titled "귀요미송."