Here are Japanese adjectives that are used to describe tastes:
amai     -   sweet

suppai - sour
shoppai - salty
karai - spicy
nigai - bitter
shibui - taste of a fruit, kaki*

umai - tastes good
mazui - tastes bad
With the taste character as a suffix, the above words become a noun for the tastes (i.e. amami, umami, etc.).  The scientific community has borrowed the word "umami" to describe the taste of MSG, which was first discovered and studied in Japan by Ajinomoto, inc.

*the fruit, kaki, has a rising tone, while the clam, kaki, has a dropping tone.  These two words almost sound the same to the untrained ear.  Shibui is a word specially reserved for this fruit, and you gotta try it to know what it tastes like.  In Japan, shibui also is a slang that means "cool".
Alternate meanings, idiomatic usages, and connotations of taste words in Japanese

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amai 甘い ("sweet") - naive, unprepared / half-baked, overly tolerant, lenient, soft, insufficient, incomplete, ineffective; succulent
This word is very commonly used to describe things other than tastes, but often has a negative connotation unlike the English usage. Occasionally used in place of "umai" to express that something is particularly delicious or in-season.

yatsu wa amai やつは甘い - he's naive / foolish. (Has a negative connotation that is very different from "A sweet person" in English.)

(person) ni amai (人)に甘い - to coddle, to be too easy on someone.

amaeru 甘える (transitive verb) - to fawn upon; to manipulate.

amai hahaoya 甘い母親 - a mother who spoils her son or daughter (negative connotation)

amaku kangaeru 甘く考える - to underestimate, to be unprepared, to think wishfully

bureeki ga amai ブレーキが甘い - the brakes are too loose / the brakes don't work well

tsume ga amai 詰めが甘い - ajar, not completely closed

amai jiru wo suu あまいじるを吸う - to take the lion's share (lit. "to drink the sweet juice")

amai sasayaki 甘いささやき - sweet nothings (very similar to English usage)

amai koe 甘い声 - sweet, seductive voice (generally has a positive connotation)

suppai 酸っぱい ("sour")
Although this word is the opposite of "amai", it has almost no idiomatic usage in Japanese. It almost always refers to things that are acidic or have a sour taste. There is, however, one commonly used proverb which uses the same kanji root:

sui mo amai mo 酸いも甘いも - good times and bad times

shoppai しょっぱい ("salty") - stingy, raspy, confused or upset
NB. In some dialects of southern Japan, the word "shiokarai" (塩辛い) or just "karai" (辛い) is used instead of "shoppai" to describe a salty taste.
In modern Japanese "shoppai" refers only to taste.
These usages of "shoppai" seem to be obsolete and not used in modern Japanese, or at least according to the people I interviewed. They are found only in literature.

shoppai koe しょっぱい声 - a raspy voice, to have a frog in one's throat (lit. "salty voice") Obs.

shoppai kao wo suru しょっぱい顔をする - to look troubled, look upset (lit. "to have a salty face") Obs.

shoppai yatsu しょっぱいやつ - a cheap bastard Obs.

karai 辛い ("spicy") - extremely severe, harsh, strict
"karai" and "tsurai" share the same writing, 辛い. "tsurai" means difficult, painful, or cruel, and some of these meanings reflect onto "karai".
NB. May also mean "salty" in dialects of southern Japan. Something with a strong minty taste is also "karai". See also karakuchi.

karakuchi no wain / sake / biiru 辛口のワイン/酒/ビール - Dry wine/sake/beer

gekikara 激辛 - extremely spicy, as in "gekikara miso ramen" - miso ramen flavoured with dried chilis and chili oil

pirikara ぴり辛 - slightly spicy, "zingy"

ano sensei wa ten ga karai あの先生は点が辛い - That teacher is an extremely strict marker.

nigai 苦い ("bitter") - difficult

nigai omoi wo saserareta 苦い思いをさせられた - I had a hard time / someone gave me a hard time (lit. "I was made to think bitterly / given bitter memories" -- a passive-causitive expression)

nigai keiken kara manabu 苦い経験から学ぶ - to learn from bitter experience

nigai koohii 苦いコーヒー - bitter coffee (NB. "shibui koohii" is never used)

nigai kao wo suru 苦い顔をする - to look bored or unimpressed (lit. "to have a bitter face")

ryouyaku wa kuchi ni nigashi 良薬は口に苦し - Good medicine tastes bitter. (saying)

shibui 渋い - "astringent" - tasteful, refined, simple, sober, suave, cool, calm, reserved, austere, elegant, traditional, unimpressed, stingy

As the respected tongpoo noted above, this taste is often associated with kaki (Japanese persimmon), to the point where Koujien defines the word as "stimulating the tounge in a way similar to the taste of a persimmon". As this fruit is not commonly eaten in most English speaking countries, both the flavour and the implied meanings are a bit difficult to describe. Apparently, the chokecherry also has this taste. Perhaps it is best described as a mix between sour and bitter, although many people would argue that it is a different taste altogether.

Esteemed TygerTyger also nodes an excellent alternative description of "shibui" as "tannic". Interestingly, English sources generally suggest that "tannic" is a sensation (or perhaps aftertaste), where in Japanese it would be normal to consider "shibui" a taste by itself.

Interestingly, most small children don't use this word and will use "nigai" (bitter) or "mazui" (yucky) instead, reflecting a change in both vocabulary and palate from child to adult. The interpretation of this word seems to vary from person to person.

shibui cha 渋い茶 - bitter tea (that there is no word in English to differentiate between "shibui cha" and "nigai cha", but an adult is more likely to say "shibui"; this does not necessarily refer to an unpleasant taste.)

shibui wain 渋いワイン - an astringent wine ("a rough wine" was also given in one dictionary, but this expression doesn't necessarily imply a bad-tasting wine)

shibui iro no kimono 渋い色の着物 - tastefully coloured / patterned kimono (actual colour would depend on the situation, wearer, and context)

shibui gara 渋い柄 - a quiet pattern (cloth, etc)

shibui kao wo suru 渋い顔をする - have a look of disgust or disapproval, make a wry face

shibui shumi 渋い趣味 - a <<cool>> hobby (expressing something that is both uncommon and likely somewhat old-fashioned, yet generally has a "cool" image; eg. kendo)

shibui haiyuu 渋い俳優 - a suave actor (almost always an older man who plays tough and gritty characters; eg. Sean Connery, Toshiro Mifune, in the noder's opinion)

umai うまい ("delicious / savoury / umami") - skilful, clever, wise, successful, fortunate

Although commonly written in hiragana, it is also possible to use the following kanji writings: 美味い, 旨い, or 巧い / 上手い. The first, using the kanji from oishii 美味しい ("delicious") specifically refers to a good taste. The second, 旨い, is the kanji generally reserved for this word and can have any of the meanings. The third (using kanji from jouzu 上手) and fourth (a rare writing using kanji from takumi 巧み) are used to refer the latter senses of doing something well or successfully. This is an example of how multiple kanji writings of the same word can be used to eliminate ambiguity in Japanese.

The word umai! うまい is often uttered by itself as an interjection -- both when tasting something delicious and when witnessing something go well, often when surprised by the outcome. When tasting food, it has a somewhat less polite sound than "oishii" 美味しい. Likewise, when delivering a compliment, it has a less polite (and often more friendly) tone than jouzu 上手.

"umai" is one of the phrases overused to the point of cliche by Japanese variety shows. Lately, its syllabic inversion, "mai'u" is sometimes used by ditzy teenagers and variety TV stars alike.

umaku iku うまく行く - to go well, succeed

nihongo umai desu ne! 日本語うまいですね - You speak Japanese very well! ( compliment given by a Japanese person to a foreigner who can speak even the meagrest Japanese)

kuchi ga umai 口がうまい - a smooth talker, silver-tongued (lit. (his/her) mouth is skilful)

umai kangae ga ukabu うまい考えが浮かぶ - to suddenly get a great idea

shiken dou datta? 試験どうだった - How was the test?
un, umaku itta to omou. うん、うまくいったと思う - Hmm, I think it went well.

Sources:
  • Koujien 5th ed.
  • Eijirou Online Dictionary, accessible at http://www.alc.co.jp/
  • Daily Concise English Dictionary 4th ed. Sanseido Co., Ltd. 1979
  • The patient assistance of many kind Japanese speakers

Shibui is perhaps more properly translated as "tannic" as the properties that persimmons, tea (n.b., 'shibui' usually describes tea that has been left to steep way too long and has become very strong, astringent, and due to the tannins in tea, tannic), and some wines share is a relatively high level of tannins. Chokecherries also have tannins, as do pomegranates and cranberries. For this reason, while we think of 'shibui' as referring to a taste, it is mostly referring to the tannic sensation. Harold McGee describes this astringency as "that dry, puckery, constricting sensation that follows on a sip of strong tea or an assertive red wine, or a bite into less than ripe fruit." (From On Food & Cooking)

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