Wait, what is a vowel?!

So here we are; I've finished my first semester as a linguistics major (ok I only took one course), but I still couldn't come up with a succinct answer when Michael Bell asked me: what's the deal with vowels? Is y a vowel or not? You know, when you're a kid, they tell you "a, e, i, o, u and sometimes y."  Why?

So I read up on it. He's right. English is one of the languages that isn't really clear about vowels.  Big surprise, right?

Phonetically speaking, vowels are sounds that are made with no constriction in the vocal tract. Basically, by pushing air from your lungs out your mouth and only shaping the sound by the way you move your tongue and lips without ever interrupting the air flow.
Phonologically speaking, vowels are the peak of a syllable.
Now here's the issue with English. We have words like little and castle, where that sound is the peak.

The issue with y and w in English, is that they meet the criteria of non-constricted sound, but they happen at the non-peak of words, too, like in yet and wet.

So we can have non-constricted sounds at the non-peak of the word.  Not vowels. 
Similarly, we have sounds at the peak of a word that are constricted sounds--also not vowels.
 
Technically speaking, a sound must match both criteria to be officially designated a vowel sound.
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This picture shows where each sound is produced.

Here is a link to a chart with audio clips of each vowel sound.  It's in the International Phonetic Alphabet, so these sounds could go for any language in the world, not just English.  The x-axis shows front, central and back--referring to where the sound is produced.  The y-axis shows close, mid and open--referring to how close to the roof of the mouth the tongue is when the sound is made.