All varieties of Braille render the characters of their respective languages in a six-dot matrix (or did until until recently); all are read from left to right, even in Hebrew; all use word-spacing, even in Chinese and Japanese; and all tend to place diacritic characters before the characters they modify.
14 0_ 0_ 00
25 __ 0_ __
36 __ __ __
EN: a b c
14 __ 0_ __ 0_ __ 00 _0 0_ _0 0_ _0 00
25 __ __ __ 0_ __ __ _0 __ _0 0_ _0 __
36 _0 __ _0 __ _0 __ 00 __ 00 __ 00 __
EN: A B C 1 2 3
= cap-a cap-b cap-c num-a num-b num-c
In English, the same formation of dots can represent either a letter or a number, depending on the preceding context. Each formation can also serve as a contraction, so that b = be, c = can, d = do, e = every, f = from, j = just, l = like, v = very, and so on.
The designers of Japanese Braille (点字) retained the letter = number equivalency, marking numbers with the same prefix, but introduced some genetic mutations to adapt to the kana syllabary. They redefined a b c d e as the vowels a i u e o, which is how everyone nowadays begins to recite the kana syllabary. The dots for these five letters are confined to positions 1-2-4 (a = 1, i = 1+2, u = 1+4, e = 1+2+4, o = 2+4), leaving positions 3-5-6 to render the consonant on each syllable, so that k = 6, s = 5+6, t = 3+5, n = 3, h = 3+6, m = 3+5+6, r = 5. The syllable n is written as m without any vowel in positions 1-2-4.
There are no capital letters in Japanese kana, but the same method is used to add the dakuten and handakuten marks to following consonants: a prefix with a dot in position 6 is used to transform h- into p-, while a prefix with a dot in position 5 is used as a to transform voiceless initials into their voiced equivalents.
14 0_ 0_ 00
25 __ 0_ __
36 00 00 00
JP: ha hi hu
14 __ 0_ __ 0_ __ 00 __ 0_ __ 0_ __ 00
25 __ __ __ 0_ __ __ _0 __ _0 0_ _0 __
36 _0 00 _0 00 _0 00 __ 00 __ __ __ __
JP: pa pi pu ba bi bu
= '-ha '-hi '-hu ''-ha ''-hi ''-hu
Braille takes up a lot of space, so its regular users rely a lot on contractions. (There's also a kind of Braille shorthand.) The word Braille itself is usually written with just the letters B-r-l. These contractions can have different meanings even in closely related members of the Braille family, like French and English. For instance, the French circumflex vowels are rendered by adding an extra dot in position six (which I will show as ^) to the first five letters of the alphabet, so â = a+^ (1+6), ê = b+^ (1+2+6), î = c+^ (1+4+6), ô = d+^ (1+4+5+6), and û = e+^ (1+5+6). (The filled dot 6 also adds a circumflex to Esperanto versions of Braille.) In English, these same contractions respectively indicate ch/child, gh, sh, th/this, and wh/which.
English double letters are contracted and rendered within a single cell by a different method: shifting the position of the dots but retaining their shape. Thus, the dots for b/but occupy positions 1+2, while bb drops to positions 2+3; c/can sits at 1+4, while cc drops to 2+5; d/do sits at 1+4+5, while dd drops to 2+5+6; and g/go sits at 1+2+4+5, while gg drops to 2+3+5+6.
A similar principle plays a key role in Korean Braille, invented in 1894 by a Canadian missionary who introduced some radical (and brilliant) mutations to adapt it to the (equally brilliant) Korean alphabet. Korean vowels occupy their own cells, while some diphthongs take up two cells. The letterㅏ(a) occupies dots 1+2+6, whileㅑ(ya) occupies its mirror image, dots 3+4+5. Similarly,ㅓ(eo) at 2+3+4 is a mirror image ofㅕ(yeo) at 1+5+6; ㅗ (o) at 1+3+6 is a mirror image ofㅛ (yo) at 4+3+6; ㅜ (u) at 1+4+3 is a mirror image ofㅠ (yu) at 1+4+6; and ㅡ (eu) at 2+4+6 is a mirror image ofㅣ(i) at 1+3+5.
The possible syllable structures of Korean are too numerous to fit into a six-dot matrix, so Korean syllables are written sequentially, typically (C)V(C), just as in French or English. In order to avoid putting spaces around each syllable, so that readers can distinguish initial from final consonants, Korean braille has two versions of every consonant, one for initial position, the other for final. Each consonant has the same shape in each position, but the one in final position is either lower than its initial counterpart or a mirror image.
Thus,ㄴ(n) occupies dots 1+4 if initial, but drops to 2+5 if final; ㄷ(d) occupies dots 2+4 if initial, but drops to 3+5 if final; andㅁ(m) occupies dots 1+5 if initial, but drops to 2+6 if final. Meanwhile, mirror-image consonants don't drop, they flip:ㄱ(g) flips from dot 4 in initial position to dot 1 in final position; whileㄹ(r) flips from dot 5 to dot 3; andㅂ(b) flips from dots 4+5 to dots 1+2. As a result, Korean 점자 'dot characters' display the same kinds of symmetry and inversion that the Korean alphabet itself displays.
Chinese Braille comes in at least two flavors, Cantonese and Mandarin. Both represent Chinese characters in three cells, one for the onset, the second for the rime, and the third for the tone, just as in Zhuyin/Bopomofo. In practice, however, tone is frequently left unmarked, generating a good deal of ambiguity. Perhaps the new system designed in the 1970s, which represents all three components in just two cells, will eventually solve that problem.
UPDATE: Matt has added a new post about attempts to render Japanese kanji in Braille. The more complicated method is geared to the shape of the kanji and requires two extra dots in each cell. The other method uses three six-dot cells per kanji. The first cell broadly classifies the type of character to follow, the second gives one mora of the Sino-Japanese reading of the character, and the third gives one mora of the native Japanese reading of the character. The second method strikes me as akin to the structural division of many written kanji into one part that broadly classifies the semantic domain, and another that indicates the (Sino-Japanese) sound value. The combination of native and Sinitic reading is also how Koreans routinely distinguish similar-sounding Chinese characters. It's as if English speakers routinely distinguished similar-sounding Latin roots by saying 'foot-ped-' vs. 'child-ped-'. The typical Japanese strategy, by contrast, is to cite a well-known compound in which the kanji occurs, just as English-speakers might distinguish 'ped- as in pedestrian' from 'ped- as in pediatrics'.