This is the official IPA chart for the Vowels.  I don't exactly subscribe to it the way it is written.

In English, we pronounce the 5 or 6 written vowels we have in a number of different ways.  But there are symbols for all of them.  But first we need to go over some background.

Vowels are produced with the tongue and lips in certain positions.  Now, when we combine two vowel signs we get what we call Diphthongs.  In some languages they combine three to get Tripthrongs.

Okay, for anyone what has studied Italian, Spanish, or Latin, the 5 normal looking vowels are pronounced the exact way as in those languages.  That means /i/ sounds like English ee (in most cases of ee), e sounds like ay (without the <y> sound), a like ah, o like oa, and u like oo. We show differences in English between what we call Tense and Lax (sometimes called long and short). 

Front Vowels:  The High Tense vowel is /i/ which sounds like the <ee> in meet, or the <ea> and <y> in easy, or most <y>s at the end of a multisyllabic word.  The lax form of that is /ɪ/.  It is called Cap i.  This is the <i> in bit, big, wick, etc.  Just below that in articulation is /e/ and /ɛ/.  The first one is a tense mid vowel that sounds like the <ay> in say, day, and in other words, only this doesn't have the /j/ sound.  Most English does have that diphthong so it's not just /e/ it's /ej/ (or /ei/) usually.  The /ɛ/ is <e> in met, rent, leg, epic, etc.  Below that is /æ/ which is the <a> in cat, had, brag, etc.

Central Vowels: There are 3 central vowels which are of any relevence to English.  There is /ə/ /ʌ/ and /a/.  /ʌ/ and /ə/ are actually pretty much the same save /ʌ/ is stressed.  I prefer to use only /ə/ so you'll have to accept that. /a/ is the <a> in Father.  It's like the ah the doctor tells you to make when he looks down your throat because he wants you to straighten you're tongue out at the bottom of your mouth.

Back Vowels:  In English these are the only rounded vowels.  Rounded means the lips move into a circular or "o" shape.  The Tense High back sound is /u/ and the lax one is /ʊ/.  /u/ is the <oo> in moon, food, and the <u> in true, and tune (not the british /ju/ pronunciation).  The /ʊ/ is used for words like book, pool, ghoul, etc.  The mid back tense vowel is /o/ and the lax is /ɔ/.  The first one, /o/ is the <oa> sound in soak, boat, road. or the <o> in wrote, smoke, etc.  The /ɔ/ is the vowel often using the <au> letter combo.  It is the vowel of all, taught, bought, etc.

Diphthongs:  The strange thing about diphthongs are they can be written a few different ways.  I prefer to use an approximate consonant (which itself is almost like a vowel which is why it is like that).  The most commonly diphthongized sounds are /i/ and /u/ which have the approximates /j/ and /w/ which sound similar.  The issue with writing them with the vowels is you don't know if they are two seperately pronounced vowels (which happens) or a Diphthong.  But expect to see a bunch of things.  The most common in English is /aj/ (written sometimes as /ai/ or /aɪ/).  In fact most of our vowels are diphthongized.  The common /e/ is usually a /ej/ (or /ei/, /eɪ/ getting the pattern?).  The vowel /o/ is usually /ow/ ( or /ou/) which why we still have words like show, grow, etc.  There is also /aw/ which is the vowel in cow, wow, ouch, etc.  In my dialect of English there is a /əj/.  This is only when you have a devoiced consonant after it, so my word <white> would be /wəjt/.

The last thing about vowels in English is that we often combine them with /ɹ/ and /l/.  The former of which is often just turned into part of the vowel and adding a little ˞ rhotic symbol.  The most common would be ɚ which sounds like the <er> in waiter.  I prefer to write them out as seperate consonants as well.  Also sometimes people like to take the /ə/ out and just use a syllabic marker underneath thusly making the sound it's own syllable.