Hepburn

A system of Japanese romanization that emphasises Western pronunciation rather than consistency or corresponding exactly to the Japanese scripts. Hepburn romanization is the most common for casual uses of Japanese, such as individual words in novels (like James Clavell's Shõgun), Japanese-English dictionaries, or anime subtitles.

Hepburn really isn't a very good system for actually studying Japanese because it makes the language look irregular and confusing where it isn't. The advantage of Hepburn, however, is that Westerners can pronounce the romanized words easily. Take the romanization of a sentence in Hepburn versus the equivalent in JSL:

Kyõ ame ga furu deshõ. (Hepburn)
Kyoo ame ga huru desyoo. (JSL)
It will probably rain today.

The Hepburn version conveys the actual pronunciation easily, whereas the JSL version is impossible to pronounce correctly for the uninitiated. Note that those ~'s over the vowels are supposed to be straight lines. Also, an important thing to remember is that Japanese vowels are essentially prounounced as they are in Spanish (this is purely coincidence).

To correctly render Hepburn Romaji in HTML, you must use the following entities. They use a macron, or line above the letter, to represent the long vowels.

Ā = Ā ā = ā
Ē = Ē ē = ē
Ī = Ī ī = ī
Ō = Ō ō = ō
Ū = Ū ū = ū

You can also use ̄ which is a combining character which will combine with any previous letter:

ā ē ī ō ū

When the macron is unavailable, the circumflex accent is usually used: â ê î ô û
Sometimes the diaeresis (also wrongly known as an umlaut) is also used: ä ë ï ö ü
Jeeves has chosen to use a tilde in his writeups, which is probably equally acceptable though I haven't seen it used this way in print.

Hepburn (Jp. hebon-shiki ヘボン式), named after its creator Dr. James Curtis Hepburn (1815-1911), is the most popular system of Japanese romanization. First used in Hepburn's 1867 Japanese-English Dictionary and codified in its present form in 1885, Hepburn is the system of choice of both the Library of Congress and Everything2. It is also occasionally known as the hyôjun (標準) or "standard" system, although the official government-recommended romanization system is not Hepburn but Kunrei. (Not that this has stopped JR and the Japanese Foreign Ministry, among others, from sticking with Hepburn.)

While slightly less regular than Kunrei system, making it more difficult for the Japanese themselves to use, it corresponds better to the sounds of English and allows the foreign non-speaker to make a decent stab at pronouncing Japanese. An actual (if extreme) example would be Hepburn jûjutsu instead of Kunrei's zyûzyutu for the martial art of 柔術. Here is a complete table for mapping hiragana to Hepburn:

あ a   か ka   さ sa   た ta   な na   は ha   ま ma  や ya  ら ra  わ wa ん n 
い i   き ki   し shi  ち chi  に ni   ひ hi   み mi         り ri  ゐ(i)
う u   く ku   す su   つ tsu  ぬ nu   ふ fu   む mu  ゆ yu  る ru
え e   け ke   せ se   て te   ね ne   へ he   め me         れ re  ゑ(e)
お o   こ ko   そ so   と to   の no   ほ ho   も mo  よ yo  ろ ro  を o 
		 
       が ga   ざ za   だ da           ば ba   ぱ pa 
       ぎ gi   じ ji   ぢ ji           び bi   ぴ pi 
       ぐ gu   ず zu   づ zu           ぶ bu   ぷ pu 
       げ ge   ぜ ze   で de           べ be   ぺ pe 
       ご go   ぞ zo   ど do           ぼ bo   ぽ po 
                    
      きゃkya しゃsha ちゃcha にゃnya ひゃhya みゃmya         りゃrya 
      きゅkyu しゅshu ちゅchu にゅnyu ひゅhyu みゅmyu         りゅryu 
      きょkyo しょsho ちょcho にょnyo ひょhyo みょmyo         りょryo 
                                   
      ぎゃgya じゃja                  びゃbya ぴゃpya 
      ぎゅgyu じゅju                  びゅbyu ぴゅpyu 
      ぎょgyo じょjo                  びょbyo ぴょpyo

NB: The characters in parentheses are obsolete.

Some additional rules worth noting:

1. Long vowels are denoted with macrons above the vowel in question, ie. o for a short vowel and ō for a long vowel. This is followed both for Japanese words and imports in katakana (e.g. byûtî "beauty"). These can also be fudged with circumflexes (like ô) since typesetting macrons can be a little hairy.
   • In Japanese words, the long vowel ee is written as ei, retaining kana spelling (example: sensei)
   • In foreign words, the long e is written ê and the long i should be written î (pâtî), although ee and ii are often seen (pâtii)
   • On E2, long vowels are completely omitted from node titles, but should be marked Hepburn-style within the content.
   
   
2. Geminate consonants (denoted in kana with a small tsu) are written by doubling the consonant in question: kokkai for 国会 "parliament", tosshin for 突進 "rush".
   • However, geminate consonants in the "ch-" series (chi, cha, chu, cho) are written "tch", eg. netchû for 熱中 "enthusiasm".
   
   
3. A syllabic n followed by a vowel should be followed by an apostrophe. This allows distinguishing between eg. kin'en 禁煙 "no smoking" and kinen 記念 "commemoration".
   
   
4. When used as grammatical particles, e is written he, ha is written wa and o is written wo. These are the sole surviving exceptions left after Japan's post-WW2 kana reform.

Modified Hepburn

These days, the standard romanization is often called "modified Hepburn" because of a single change: according to original Hepburn, the combinations 'np', 'nb', and 'nm' should be written 'mp', 'mb' and 'mm' respectively. This is phonetically more accurate but obscures the original syllables, so the modern convention is to retain the n and write eg. kinpatsu and shinbun instead of kimpatsu and shimbun for 金髪 "blonde" and 新聞 "newspaper". This change is relatively recent (the Library of Congress switched in 1983) and the older forms are still often seen, especially in better-known place names like Namba and Gumma.

References

sci.lang.japan AFAQ 5.3.2
www.library.ucla.edu/libraries/eastasian/jpntable.htm
Thanks to Cogito and bakufu for comments