The Japanese hiragana and katakana syllabaries can mostly be described as
phonetic. But there are two exceptions, the two pairs of syllables modified to
be voiced with the _dakuten_ diacritic which turns them into homophones:

  * す (su) → ず (zu); つ (tsu) → づ (zu) 
  * し (shi) → じ (ji); ち (chi) → ぢ (ji)

The same goes for katakana:

  * ス (su) → ズ (zu); ツ (tsu) → ヅ (zu)
  * シ (shi) → ジ (ji); チ (chi) → ヂ (zi)

So are these pairs always pronounced the same or are there sometimes subtle
differences? The former of each pair seems to be the more common, are the
latter in each pair much used? How does one choose which to use when spelling
words in kana? What about for spelling foreign words or names in kana?
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question
pronunciation orthography phonology kana kana-usage
What are the differences between じ and ぢ, and ず and づ?
In modern Japanese these pairs are pronounced exactly the same:

  * ず, づ are pronounced either `[dzu]` or `[zu]`. 
  * じ, ぢ are pronounced either `[dʑi]` or `[ʑi]`.  
(the first sounding like the English J and the second like the French J, but
both are with the middle of the tongue raised to the hard palate, producing
what seems like a softer pronunciation).

So in short, the pairs are redundant as far as the modern language is
concerned. But like most cases of duplicate letters, they originally
represented distinct sounds:

  * ず represented `[zu]` (nowadays it is really more like `[dzu]` with `[zu]` being a slacker pronunciation).
  * づ represented `[du]` (and just like `[tu]` became `[tsu]` it became `[dzu]`)
  * じ represented `[zi]` (and just like `[si]` became `[ɕi]` it became `[ʑi]`).
  * ぢ represented `[di]` (and just like `[ti]` became `[tɕi]` it became `[dʑi]`).

While the exact pronunciation of these 4 letters have changed since classical
Japanese, they essentially remained distinct until the Edo period. We can
easily see this from old Portuguese transliterations of Japanese, which used
the following letters to transliterate these sounds:

  * じ was transliterated `ji`.  
(remember that this is the Portuguese or French `j` here, not the English
`j`).

  * ぢ was transliterated `gi`.  
(it was obviously distinct from ぎ, but not distinct enough to Portuguese or
Spanish ears)

  * ず was transliterated `zu`.
  * づ was transliterated `zzu` or `dzu`  
I'm not sure how this last letter was pronounced, since the only old text I've
read, _Ars Grammaticae Japonica_ , makes a clear distinction between 水 (old
spelling: みづ) which it transliterates as _mizzu_ and 蜜 (strangely enough: みつ)
which it spells as _mizzu_. This mess probably stems from the fact that
Spanish itself (the writer is Spanish) was undergoing transformations to the
very same consonants at that time.

In the end, `[dzu]` merged with `[zu]` and `[dʑi]` merged with `[ʑi]` in all
but a few dialects (the Kagoshima dialect apparently retains these
distinctions), but until the spelling reforms of 1946 all words retained their
original spellings, so 水 was spelling ミヅ in and 味 was spelled アヂ. Good kokugo
(Japanese-Japanese) dictionaries still list these old spellings.

After the spelling reforms, づ and ぢ were kept in only two places:

  1. With a repeated sound: [続く]{つづく}, [縮む]{ちぢむ}
  2. In compounds with [rendaku](http://en.wikipedia.org/wiki/Rendaku): [気付く]{きづく}, [馬鹿力]{ばかぢから}

In all of the other places, ぢ was replaced with じ and づ with ず.
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answer
Pronunciation-wise, there is no difference in the standard dialect. Some
dialects may preserve the distinction between the two sounds, but most of the
words that used to be spelled with づ and ぢ are now spelled with ず and じ in the
standard language. (In other words, relying on the standard spelling won't
tell you when to use "dzu" and "dji" in these dialects, because standard
spelling doesn't reflect when these sounds occur.) [This
page](http://en.wikipedia.org/wiki/Yotsugana) has more information on
dialectical usage of づ and ぢ.

The cases where づ and ぢ are still used today are when rendaku is applied.
"Rendaku" is the voicing of syllables that sometimes occurs when making
compound words. For example, the same phonetic process that turns 山口 into やまぐち
(instead of やまくち) also turns 月々 into つきづき. One of the most common cases is the
verb つく (付く), which gets tacked onto other verbs: もと+つく = もとづく. つける can turn
into づける via the same process.

Also, occasionally, the sequence "tsuzu" is spelled つづ. For example, 続ける is
つづける even though it doesn't seem to be a compound word.

ぢ is considerably rarer than づ for some reason -- I can't think of any words
offhand that use it, though I may know one or two -- but nonetheless the same
rules apply.
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answer
Here's what my Japanese lecturer told me when I asked her about it:

"Usually it is じ for ji sound. However, when ji is used after chi sound in one
word with one kanji, ぢ is used, such as, ちぢむ （縮む）、ちぢれる（縮れる）. When it is a part
of word with two kanji, such as, ちじん （知人 = acquaintance), じ is used."

Interesting....
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With the writing reform almost most instances of the old ぢ・ヂ and づ・ヅ have been
replaced with the homophonic じ・ジ and ず・ズ with the following exceptions:

  1. ぢ・ヂ and づ・ヅ are still used in words containing a voiced repeated ち or つ (i.e. one that could be written with a voiced iteration mark ゞ), e.g.

> ちぢむ（縮む） つづく（続く）

  2. ぢ・ヂ and づ・ヅ may appear as a result of _rendaku_ (when the first _kana_ of a compound word is voiced), e.g.

> はなぢ（鼻血）

For reference, the relevant section of the [cabinet
announcement](http://www.mext.go.jp/b_menu/hakusho/nc/t19860701002/t19860701002.html)
by the _Ministry of Education, Culture, Sports, Science and Technology_
(文部科学省), containing all common examples, is

> 5. 次のような語は、「ぢ」「づ」を用いて書く。
>
>   1. 同音の連呼によって生じた「ぢ」「づ」
>
> **例** ちぢみ(縮) ちぢむ ちぢれる ちぢこまる
>
> つづみ(鼓) つづら つづく(続) つづめる(約△) つづる(綴＊)
>
> 〔注意〕 「いちじく」「いちじるしい」は、この例にあたらない。
>
>   2. 二語の連合によって生じた「ぢ」「づ」
>
> **例** はなぢ(鼻血) そえぢ(添乳) もらいぢち そこぢから(底力)
>
> ひぢりめん いれぢえ(入知恵) ちゃのみぢゃわん
>
> まぢか(間近) こぢんまり
>
> ちかぢか(近々) ちりぢり
>
> みかづき(三日月) たけづつ(竹筒) たづな(手綱) ともづな にいづま(新妻) けづめ ひづめ ひげづら
>
> おこづかい(小遣) あいそづかし わしづかみ こころづくし(心尽) てづくり(手作) こづつみ(小包) ことづて はこづめ(箱詰) はたらきづめ
> みちづれ(道連)
>
> かたづく こづく(小突) どくづく もとづく うらづける
>
> ゆきづまる ねばりづよい
>
> つねづね(常々) つくづく つれづれ
>
>

>
>
> なお、次のような語については、現代語の意識では一般に二語に分解しにくいもの等として、それぞれ「じ」「ず」を用いて書くことを本則とし、「せかいぢゅう」「いなづま」のように「ぢ」「づ」を用いて書くこともできるものとする。
>
> **例** せかいじゅう(世界中)
>
> いなずま(稲妻) かたず(固唾＊) きずな(絆＊) さかずき(杯) ときわず ほおずき みみずく
>
> うなずく おとずれる(訪) かしずく つまずく ぬかずく
>
> ひざまずく あせみずく くんずほぐれつ さしずめ でずっぱり なかんずく うでずく くろずくめ ひとりずつ
>
> ゆうずう(融通)
>
> 〔注意〕
> 次のような語の中の「じ」「ず」は、漢字の音読みでもともと濁っているものであって、上記(1)、(2)のいずれにもあたらず、「じ」「ず」を用いて書く。
>
> **例** じめん(地面) ぬのじ(布地) ずが(図画) りゃくず(略図)
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