Romanized Korean lyrics are sometimes refered to as Konglish, but that's also the term used for baffled Korean-English (such as Spanglish, Engrish, Franglais, etc.). Generally speaking, you would just say "Romanization".

As far as I know, there are no computerized Hangul Romanization engines. Furthermore, Korean Romanizations are generally terrible and nonuniform, because of the language's difficult-to-express-in-Roman-characters phonology. If you have a song in particular, I'd be willing to Romanize it for you.

But honestly, the best way would be to simply learn Hangul; it's just one of those languages that doesn't make sense until you look at it the way it's meant to be looked at. The good news is that Hangul is one of the most efficient and easy-to-learn/understand alphabets in the entire history of mankind. It was created by scholars and scientists under the rule of Great King Sejeong, and is such a celebrated part of Korean culture that it even has its own national holiday. I LOVE Hangul; it can really be learned in a few hours. I suggest looking Hangul up on Wikipedia, though; there's a great article.

For now, here is a crash course in Hangul, using a modified form of the official South Korean Romanization system. (Note: there are many, many others... this is actually my least favorite, but the most widely used).

ㄱ K; G (before vowel)
ㄲ KK
ㄴ N
ㄷ T; D (before vowel)
ㄸ TT
ㄹ R; L (followed by consonant)
ㅁ M
ㅂ P; B (before vowel)
ㅃ PP
ㅅ S
ㅆ SS
ㅇ silent (beginning of syllable); NG (end of syllable)
ㅈ CH; J (before vowel)
ㅉ CCH
ㅊ CH'
ㅋ K'
ㅌ T'
ㅍ P'
ㅎ H
ㅏ A
ㅐ AE
ㅑ YA
ㅒ YAE
ㅓ EO
ㅔ E
ㅕ YEO
ㅖ YE
ㅗ O
ㅘ WA
ㅙ WAE
ㅚ OE
ㅛ YO
ㅜ U
ㅝ WEO
ㅞ WE
ㅟ WI
ㅠ YU
ㅡ EU
ㅢ UI
ㅣ I

The pronunciation of a letter depends on the sounds before and after it. The best illustration of this is the names of the letters themselves. Note the difference between the way the letter is spoken at the beginning of a syllable, versus the end of a syllable.

[Taken from Wikipedia]

ㄱ gi-yeok (기역), or gi-euk (기윽) in North Korea
ㄴ ni-eun (니은)
ㄷ di-geut (디귿), or di-eut (디읃) in North Korea
ㄹ ri-eul (리을)
ㅁ mi-eum (미음)
ㅂ bi-eup (비읍)
ㅅ si-ot (시옷), or si-eut (시읏) in North Korea
ㅇ i-eung (이응)
ㅈ ji-eut (지읒)
ㅊ ch'i-eut (치읓)
ㅋ k'i-euk (키읔)
ㅌ t'i-eut (티읕)
ㅍ p'i-eup (피읖)
ㅎ hi-eut (히읗)

You can probably tell by now that Hangul is written in blocks, each block being one full syllable. Here are the Hangul and Romanized lyrics to BoA's "We".

어둠이 날 덮고 하늘에 달 잠겨도
Eodum-i nal deopgo haneur-e dal jamgyeodo
내 곁에 너만 있다면 세상은 내겐 눈부셔
Nae gyeot'-e neo-man ittdamyeon sesang-eun naegen nunbusyeo

부디 부디 어떤 의미라도 함께 한다면
Budi budi eotteon uimi-rado hamkke handamyeon
하루 하루 너의 삶에 내가 여백되어 살텐데
Haru haru neo-ui salm-e nae-ga yeobaekdoe'eo salt'ende

니 곁에 함께할 수 있다면 무엇도 하나 두려울게 없는 나
Ni gyeot'-e hamkkehal su ittdamyeon mu'eotdo hana duryeo'ul ge eobtneun na
세상에 니가 있을 곳은 바로 여긴데 돌아와 어서 내 곁으로
Sesang-e ni-ga isseul gos-eun baro yeoginde dorawa eoseo nae gyeot'-euro

죽어도 좋겠어 널 위한 이유라면
Jugeodo johgesseo neor wihan iyu-ramyeon
그대로 너의 맘 깊이 영원히 살 수 있으니
Geudae-ro neo-ui mam gip'i yeongweonhi sal su isseuni

가도 가도 끝이 없는 거친 길을 걸어도
Gado gado kkeut'-i eobtneun geoch'in gir-eul georeodo
하루 하루 험한 세상 속에 내가 던져 진대도
Haru haru heomhan sesang sog-e nae-ga deonjyeo jindaedo

니 곁에 함께할 수 있다면 무엇도 하나 두려울게 없는 나
Ni gyeot'-e hamkkehal su ittdamyeon mu'eotdo hana duryeo'ul ge eobtneun na
세상에 니가 있을 곳은 바로 여긴데 돌아와 어서 내 곁으로
Sesang-e ni-ga isseul gos-eun baro yeoginde dorawa eoseo nae gyeot'-euro

넌 내게 또 하나의 나 인걸 내 안엔 니가 숨을 쉬고 있는데
Neon naege tto hana-ui na in'geol nae an-en ni-ga sum-eul swigo ittneunde
이 세상 같은 하늘아래 살아 있다면 돌아와 어서 내 곁으로
I sesang gat'eun haneur-arae sara ittdamyeon dorawa eoseo nae gyeot'-euro

It is important to remember, though, that orthography (writing) and phonology (pronunciation) are two entirely separate things, and things are often pronounced differently than they appear. For example, 없는 looks like it would be pronounced "eobtneun", while it actually sounds more like "eomneun". 눈부셔 looks like "nunbusyeo", but it actually sounds more like "nunbusheo". If you get deep into Korean phonology, you will see that this makes a lot of sense because of the way the human mouth works. But for now, transliterating each individual letter will at least give you a basic idea of how the word might sound. (Listening to the singer is an even better way!) This is why Romanizing Hangul is such dirty business, and why it is best (and easier, in the long run) to simply learn how to read the raw Korean.