Vowel sounds can be visualized in a two-dimensional space according to tongue position. The vertical axis is runs from open down to closed, and the horizontal runs from front to back. See a linguistics textbook for far more detail.
English has five vowel letters, but a lot more than five vowel sounds. Scholars argue about how many vowel sounds English and other languages have because there’s room for disagreement on how much two sounds can differ and still be considered variations on the same sound. The IPA Handbook  lists 11 vowel sounds in American English, not counting diphthongs.
Click through for comparative charts and some bonus expertise in the comments.