One meaning of this term is its use in phonology to refer to one of two types of speech sound: vowels and consonants (See Consonant). Unlike consonants, vowels are produced without the speech organs (See Articulators/ Speech Organs) blocking the outgoing air. There are 20 vowels in RP English including both single vowels and diphthongs. In this meaning, vowel is sometimes called vowel sound.
Another meaning is the written symbol used to represent a vowel. In English these are: a, e, i, o, u, and sometimes y.
I think it is very useful to use the phonemic chart to teach English vowel sounds bit by bit. I think it really helps learners, particularly older ones, to hear the difference between the vowel sounds and get a feel for where and how to pronounce them.
Dalton, C. and Seidlhofer, B. (2004) Pronunciation. Oxford: Oxford University Press.
Kenworthy, J. (1987). Teaching English Pronunciation. Harlow: Longman.
Thornbury, S. (2006). An A-Z of ELT. Oxford: Macmillan.
Underhill, A. (2005) Sound Foundations. Oxford: MacMillan.