The second reason vowel sounds present difficulties is that we do not have letter-sound correspondence in English. That is to say, one letter does not always represent the same sound. So the written letters A, E, I, O and U do not correspond to just one vowel sound. In addition, the same vowel sound can have multiple spellings, depending on the word context. (More information on vowel sounds and spelling can be found here.)
So, given that, how do we know what sound any given written vowel will make in a word? A great way to answer this question is to use an online dictionary, which will allow you to listen to the word as well as view the IPA (International Phonetic Alphabet), which has symbols representing the sounds. Let's take a closer look at the letter A and the vowel sounds it can represent.
Vowel a /æ/ as in at, man, and bad is a frequently mispronounced vowel in American English. Error patterns tend to be either that we use vowel ay /eɪ/ as in gate or vowel eh /ɛ/ as in get.
Vowel aw /ɔ/ as in awesome, ball, and taught can sometimes be confused with with vowel ah /a/ as in hot. Vowel aw has lip rounding and vowel ah does not. The contrast can be heard in the words dawn (sunrise) and don (to put on) or yawn (deep breath) and yon (as in yonder). Some American dialects, such as that spoken in the Pacific Northwest do not differentiate these two vowels.
Vowel ah /ɑ/ (rounded) as in want, water, and wash is the almost the same vowel as vowel ah /a/ (unrounded) which is spelled with letter o, as in hot and not. The difference is that the round lips of the w sound change the vowel to be rounded as well.
Vowel Schwa /ə/ as in across, about, and extra is the sound we use for some unstressed syllables. The most common error on vowel schwa tends to be using vowel ay because of the letter ay. So you might hear someone pronounce about or across as ay-bout or ay-cross. This in incorrect and the vowel should sound more like "uh." Uh-bout, uh-cross.
To hear and see these sounds produced, take a look at the video below.