A vowel is a particular kind of speech sound made by changing the shape of the upper vocal tract, or the area in the mouth above the tongue. In English it is important to know that there is a difference between a vowel sound and a [letter] in the [alphabet]. In English there are five vowel letters in the alphabet.

The sounds of American English are written with letters in the English alphabet, as either vowels or consonants. All English words are written with vowel letters in them.

These letters are vowels in English:
A, E, I, O, U, and sometimes Y.

It is said that Y is "sometimes" a vowel, because the letter Y represents both vowel and consonant sounds. In the words cry, sky, fly, my and why, letter Y represents the vowel sound /aɪ/. In words like myth and synchronize, Y represents the vowel sound /ɪ/. In words like only, quickly, and folly, Y represents the vowel sound /i/.

It can also be a consonant sound called a glide as in the beginning of these words: yellow, yacht, yam, yesterday. Y is a consonant about 2.5% of the time, and a vowel about 97.5% of the time.

The letter W can sometimes be the second part of a vowel sound as in words like such as cow, bow, or how. In these words the vowel has the sound of /aʊ/. The letter W can be used as a consonant sound at the beginning of in the words when, where, wet. In some languages, like Welsh, the letter W represents the vowel sound /ʊ/, like cwm (a kind of valley).

In written English the six vowel letters are used to represent the 13-15 vowel sounds (depending on the variety) in English. This means there are many more vowel sounds than letters in the English alphabet, and the English spelling systems doesn't always help us figure out what the English sounds are. This can be confusing.
 The rest of the letters of the alphabet are consonants:
B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, W, X, Y, and Z.

Monophthongs and diphthongs 
Simple vowels are called monophthongs. The letters, like /ɪ/, are the IPA letters for each vowel sound in English. (The IPA is the International Phonetic Alphabet). In the IPA, each symbol represents a different sound, so using the IPA is helpful in pronouncing words.

Common monophthongs in English (these are for General American English) include:
 /i/ as in police, feet, eat, and silly
 /ɪ/ as in it, sit, kick, myth and bitter
 /ɛ/ as in end, bet, less, and letter
 /æ/ as in at, apple, fat, and matter
 /u/ as in cool, tune, soup, and kung fu, 
 /ʊ/ as in cook, should, pudding, foot, and rook
 /ʌ/ as in bus, blood, come, and up
 /ə/ as in kingdom, photography, philosophy, ketchup, and hundred
 /ɚ/ as in butter, collar, flavor, firm, and burst
 /ɔ/ as in all, fought, hot, and bot
 /ɑ/ as in father, walk, arm, heart, wasp, lager, envelope and aardvark

Diphthongs are a combination of two different vowel sounds, one vowel sounds turns into another sound as you say them. If you pronounce the words below slowly, you can hear the two vowel sounds of the diphthongs.

Common diphthongs in English include:
 /eɪ/ as in ate, reign, vain, flavor, slay, and convey
 /oʊ/ as in toe, row, go, boat, mode, and chateau
 /aɪ/ as in eye, I, pie, cry, cypher, climb, lime, light, kayak, Thai, and height
 /aʊ/ as in loud, house, cow, about, Daoism, and Macau
 /oɪ/ as in boy, moist, and Freud

Like other languages, there are many dialects of English, and different dialects often use different vowel sounds. But the IPA symbols can tell us which vowel sound a dialects uses. For example, some American English speakers differentiate between the vowels in the words cot and caught, while in other dialects these words are homophones. People who study the differences between the dialects of English often study the different way vowel sounds are pronounced.

The difference between the way English is spelled and the way the words are pronounced came about because all languages change, so spoken English changes, but the spelling system does not.

The study of speech sounds is called phonetics.

Related pages 
 Orthography
 Dialect
 Diphthong

References 

Linguistics