I cooled Webster 1913's definition (see below) to answer several burning questions that leapt out at me today:
One reason questions such as these arise is that English spelling isn't strictly phonetic (I know, thank you Captain Obvious), and besides that there's exceptions to the rules many of us learned in phonics back in the day. In phonetic transcription, many of the examples given in When is W a vowel? don't contain a /w/ (although that symbol is sometimes used to indicate a diphthong with final rounding). Likewise, the "y" in "sky" and "crypt" do not represent the same sounds, although both are vowels. The "y" in "you" is written /j/ in IPA, and diphthongs such as the one in "say" may be written with a /j/, /y/ or /I/ as their second vowel (which is actually a semivowel or glide, as some phoneticians term them). But I digress. See Gritchka's excellent writeup vowel for the phonetic features that make a vowel a vowel (what Webby terms the "nature" of the sound).