"row-MON-ghoul" - a portmanteau word combining "romanize" (to convert a non-roman alphabet into approximate equivalent letters of the roman alphabet) and "hangul" (the language of the Koreas), to mean "words converted from hangul symbols into roman letters."
For instance the phrase 소녀시대 would offer little or no indication of pronunciation to speakers of Romance languages (English, French, Italian, etc), who have no training in Hangul/Korean. However, in romangul, the phrase would be Sonyeo Sidae, or So Nyuh Shi Dae, which offers English-speakers at least an approximate understanding of proper pronunciation.
Romangul is especially useful for fans of Korean music wishing to sing along with the music in that language, without knowing the language itself, per se -- or students of the language attempting to memorize corresponding Hangul characters with their appropriate sounds.
Similarly, romaji is the combination of "romanize" with "kanji" (the written language of Japan). Romangul is to Korean, as Romaji is to Japanese.
I can't tell what particular sounds they're using just from the audio, and the Hangul version of the lyrics don't help because I don't know Korean. Can someone write the lyrics in Romangul for me, so I can know how to pronounce them properly?