The length of a vowel is in physical terms simply its duration. Languages exploit the possibilities of vowel length in different ways. We must first distinguish uses of duration: you can draw out the vowel of looong for effect. Probably most languages allow this. It is not, however, part of the language proper. In contrast, the length difference between sit and seat is part of a phonemic contrast: it makes a difference between words.

Quite a lot of languages have no such contrast. In them any vowel is about as long as any other. These include Chinese, Modern Greek, Polish, and Spanish.

At the other end are languages that have contrasting long and short versions of each vowel: so in these [kat] and [kaat] would be different words, pronounced the same except for the duration of the vowel. Languages of this type include Finnish, Japanese, Czech, and Arabic.

notation

The phonetic symbol for length is a colon made of two small triangles pointed to each other. I show this here as an ordinary colon, thus [ka:t]. Length is variously represented in scripts: in Welsh the circumflex is used (only on some words), in Czech and Hungarian the acute, and in Finnish and Somali the vowel is written twice. In Japanese it is either written with a repeated vowel (in hiragana) or with a stroke (in katakana). In Arabic only long vowels are written, using a consonant that is related to the vowel. In general it doesn't matter whether length is shown by doubling or by a length mark. No language that I know of consistently distinguishes words such as [kaat] versus [ka:t]. However, in marginal cases there can be a difference, where the two vowels belong to different syllables: e.g. English carrying may have a sequence of the form [ii] whereas careen has [i:]; and consonant mutation in Finnish produces raa'an from raaka.

patterns of long vs short

In between the all-or-nothing examples I've mentioned so far, there are often other arrangements of long versus short vowels. In English, Hungarian, and Latin, both long and short vowels occur but they do not precisely correspond with each other. In Ancient Greek there were seven long but only five short vowels. In French, Italian, and Scottish English long vowels occur in a narrow range of positions and in general do not affect meaning.

In Classical Latin there were five short vowels [a e i o u] (which is why our alphabet has just those vowel symbols), and five long vowels [a: e: i: o: u:]. Classical Latin changed into Vulgar Latin, the most recent ancestor of the Romance languages, and the sound changes between Classical and Vulgar give us clear evidence of the different sounds: both short [i] and long [e:] of CL became VL [e], whereas CL short [e] became a more open sound [ε] in VL, while long [i:] became [i]. Vulgar Latin had no length distinctions. This shows us that, for example, [i:] and [i] were as in English seat, sit: similar but not identical. (For the record, here are their exact values in full, using SAMPA notation: [a E I O U a: e: i: o: u:].)

In Hungarian we get a similar thing: the seven short vowels [a e i o ø y] (written a e i o ö u ü) differ in quality as well as length from their long counterparts. (Those exact values in full: [Q æ I O œ U Y a: e: i: o: ø: u: y:].)

English

English is like Latin and Hungarian in having long and short vowels that don't exactly correspond, but in the other two we can clearly match the vowels as pairs, slightly differing in quality. In English the pairing is not as neat. It also differs by dialect. Scottish English is a special case, to be discussed below. American English lacks a short [o]. Depending on the dialect it may be debatable which short vowel pairs with which long vowel; indeed many modern descriptions of English don't show vowel length, but show the quality only: they use the symbol [I] in sit and [i] in seat, and note that [i] is normally longer than [I].

This notation may be misleading: you sometimes see it claimed that English 'has no vowel length', or has no contrastive or distinctive vowel length. This is only true under a narrow interpretation. Except for Scottish, we could notate sit and seat as [sit] and [si:t] and note that short [i] is more precisely [I]. What English has is [I] versus [i:], contrasting both quality and length together. It is rare for any English dialect to make any contrast of purely length. (In my accent the only common contrast of this type is [kat] cut versus [ka:t] cart.)

An important reason why the [sit ~ si:t] notation is no longer used is that it is not actually duration that causes the contrast. If you lengthen the vowel in sit it still sounds like sit, and conversely a short seat is still identifiable as seat, not sit. The quality of the vowel is what's important, not the length. In the early days of phonetics this was described as height (of the tongue in the mouth); later the vague terms tense and lax were used; finally in the 1960s it was discovered that the essential difference is the frontness or backness of the tongue root: [i] has advanced tongue root and [I] lacks it.

Another reason why [i ~ i:] is not an adequate description of the length difference is that in English the following consonant systematically affects length. Voiceless sounds such as [t k p s f] shorten the vowel quite a lot, so sit is considerably shorter than Sid in normal speech, and seat is shorter than seed. There is no definite relation between the Sid and seat vowels, but they are typically about equal, giving three grades of duration: [sIt ~ sId,si:t ~ si:d].

In most dialects of English the long vowels are not 'pure', but are more a diphthong, and [i:] is more accurately something like [Ii] or even [əI]. The vowel of mate is pure [e:] or diphthong [eI] depending on dialect. This is another reason why it's not obvious that we should or can pair long vowels against short ones in English.

To confuse matters there is a traditional terminology of 'long vowels' and 'short vowels' in English, under which mate, mete, mite, mote, mute have long vowels corresponding to the short ones of mat, met, nit, not, nut, and in dictionaries their pronunciation would be marked with the breve and macron: măt, māte. In Middle English these actually were long/short pairs (and that's what the silent -e indicated), until the Great Vowel Shift realigned them. The terminology has remained, but is misleading compared to a phonetic definition of length. The shift gave us a large residue of pairs of related words, where one was shortened or lengthened by a Middle English rule, where this traditional short/long distinction still holds: such as child/children, keep/kept, divine/divinity, opaque/opacity. Some linguists (notably Chomsky) believe we can regard this alternation as a still-living feature of modern English, and e.g. keep ~ kept is stored by present-day speakers as [ke:p ~ kept] rather than [ki:p ~ kept]. Personally I find this unconvincing, and I think we use analogy more than rules when we relate such pairs. However, it is true that that old distinction is well entrenched in our grammar, whereas nothing much systematically connects phonetically close pairs such as [I ~ i:].

restricted occurrence

In Italian long vowels only occur stressed, in an open syllable. So [kane] (open [ka]) would be pronounced [ka:ne], whereas in [kanna] (closed [kan]) the stressed [a] remains short. This is not contrastive: any vowel in this position is lengthened, and no others are.

In French length also occurs only under stress. The final syllable of a phrase is stressed, and this is long if it's a closed syllable ending in certain consonants ([v r z Z]), or closed and containing certain vowels (including nasals) -- I won't give more precise detail, as it's likely to vary a bit with age and dialect. Mostly length is automatic, not contrastive, but for some speakers there is or was a phonemic contrast between [mεtr] mettre, mètre and [mε:tr] maître.

In Scottish accents of English there are no length distinctions of the sit ~ seat type, the contrast being carried purely by the quality: [sIt ~ sit]. In the standard accent final stressed vowels are long, as in [si:] see, sea. This length is retained when inflection is added. This can create a pure length contrast: [siz] seize vs [si:z] sees, seas; and likewise [brud] brood vs [bru:d] brewed. I have noticed that some accents (Glasgow, I think) lengthen stressed vowels.

how long?

A long vowel in a two-way contrast is typically one and a half times to twice the length of a short vowel. Some languages have intermediate-length vowels in certain positions, such as "short" vowels under stress, or perhaps unstressed "long" vowels; however these are never contrastive, but are allophonic variations, like the English three-way system. While two-way contrasts are common (e.g. [kat] and [kaat] mean different things), the only language commonly cited as having a three-way contrast (e.g. [kat ~ kaat ~ kaaat] all with different meanings) is Estonian, and even there the analysis is disputed. Some phoneticians say the three grades of length can be accounted for by other features, as they can in English.

Traditional guides to the pronunciation of Biblical Hebrew include extra-short vowels (the hateph vowels) beside ordinary short and long vowels. This is almost certainly wrong, since such contrasts are not known to occur in any living language. It was probably an optional choice between full vowel or schwa.