DIPHTHONGS (δίφθογγοι)

The vowels ι and υ can be joined to the other vowels to make one syllable.  These diphthongs are:

αιas in “aisle”
αυas in “hour”
ειas in “eight”
ευthe vowels of “met you”
ηυthe vowels of “hey you”
οιas in “oil”
ουas in “group"
υιlike “we”

When ι is added to η, ω, or a long α, it is not pronounced.  Unless the first vowel is capitalized, the ι is written beneath it (ᾳ, ῃ, ῳ) and is called “iota subscript.”  Pretend it isn’t there when you say the word.

Accents and breathings are placed over the second vowel of a diphthong, except for iota subscript or equivalent.

A diaeresis (two little dots, αϊ) indicates that two vowels are to be pronounced separately, not as a diphthong.

Some examples:
αὐτόςself
(note the breathing over the υ, the second letter of the diphthong.  This is thus a two-syllable word even though it has three vowels.)
ΝεῖλοςNile
πνεῦμαbreath
ΘρᾴκηThrace (ignore the iota subscript in pronunciation)
ΕὐκλείδηςEuclid
ἏιδηςHades
(a tricky one; since the accent and breathing are in front of the Α, Αι is not a diphthong in this word, and thus the ι is not pronounced.  If the word were not capitalized, the iota would be a subscript: ᾇδης)