Romanization of Korean refers to systems for representing the Korean language in the Latin script. Korea's alphabetic script, called Hangul, has historically been used in conjunction with Hanja (Chinese characters), though such practice has become infrequent.
Romaja literally means Roman letters in Korean, and refers to the Latin script. "Romaja" is not to be confused with "romanization". The former can be applied to any use of the Latin script in Korean text—whether for Korean or non-Korean words or names—while the latter refers to writing Korean words using the Latin script: either romanizing individual words in a Korean text, or writing an entire Korean text in the Latin script.
Systems
Many romanization schemes are in common use: Revised Romanization of Korean (RR, also called South Korean or Ministry of Culture (MC) 2000): This is the most commonly used and widely accepted system of romanization for Korean. It includes rules both for transcription and for transliteration. South Korea now officially uses this system which was approved in 2000. Road signs and textbooks were required to follow these rules as soon as possible, at a cost estimated by the government to be at least US$500–600 million. Almost all road signs, names of railway and subway stations on line maps and signs etc. have been changed. Romanization of surnames and existing companies' names (e.g. Hyundai) has been left untouched; the government recommends using the new system for given names and new companies. RR is similar to MR, but uses neither diacritics nor apostrophes, which has helped it to gain widespread acceptance on the Internet. In cases of ambiguity, a syllable boundary may be indicated with a hyphen, but this is truly optional; it is not mandatory. Thus, state institutions usually do not make use of this option. McCune–Reischauer (MR; 1939): the first transcription to gain some acceptance. A slightly modified version of MR was the official system for Korean in South Korea from 1984 to 2000, and yet a different modification is still the official system in North Korea. MR uses breves, apostrophes and diereses, the latter two indicating syllable boundaries in cases that would otherwise be ambiguous. Several variants of MR, often also called "McCune's and Reischauer's", differ from the original mostly in whether word endings are separated from the stem by a space, by a hyphen or not at all; and if a hyphen or space is used, whether sound change is reflected in a stem's last and an ending's first consonant letter (e.g. pur-i vs. pul-i). Although mostly irrelevant when transcribing uninflected words, these variants are so widespread that any mention of "McCune–Reischauer romanization" may not necessarily refer to the original system as published in the 1930s. MR-based romanizations have been common in popular literature until 2000. The ALA-LC / U.S. Library of Congress system is based on but deviates from MR. Unlike in MR, it addresses word division in seven pages of detail. Syllables of given names are always separated with a hyphen, which is expressly never done by MR. Sound changes are ignored more often than in MR. ALA-LC also distinguishes between ‘ and ’. Yale (1942): This system has become the established standard romanization for Korean among linguists. Vowel length in old or dialectal pronunciation is indicated by a macron. In cases that would otherwise be ambiguous, orthographic syllable boundaries are indicated with a period. This system also indicates consonants that have disappeared from a word's South Korean orthography and standard pronunciation. ISO/TR 11941 (1996): This actually is two different standards under one name: one for North Korea (DPRK) and the other for South Korea (ROK). The initial submission to the ISO was based heavily on Yale and was a joint effort between both states, but they could not agree on the final draft. A superficial comparison between the two is available here: Lukoff romanization, developed 1945–47 for Fred Lukoff's Spoken Korean coursebooks Romanization of Korean (1992): The official romanization in North Korea. This is also based on but deviates from MR.