There are many more to this but I will try to sum up the basic ones:
Augment happens when the verb begins with a consonant and augments the "root/theme" (i don't know the word in english-I mean the part of the verb without the ending) to two syllabes, so it augments by one syllable the one-syllable (only) verb themes (π.χ. έ-λυν-α, έ-λυσ-α, έ-γραφ-α, έ-γραψ-α), because past tenses are always accented on the antepenult. When the verb "theme" has more than one syllables, augment does not occur (because there is no need for an antepenult-there is already one). (π.χ. γέ/λασ-α, μί/λησ-α, ζω/γρά/φισ-α). So augment may occur in some persons only: for example έ-πια-σα but πιά-να-με (because there is already an antepenult)
Some irregular verbs are not augmented, such as: βγαίνω - βγήκα, βρίσκω - βρήκα, μπαίνω - μπήκα, πηγαίνω - πήγα, παίρνω - πήρα. etc
In some other irregular verbs, the augment is an η- not an ε-, such as: θέλω-past cont: ήθελα, ξέρω-past cont. ήξερα, πίνω-past simple: ήπια.
Also, augment occurs in compound verbs before the part of the word that is actually the verb for example διαμένω=δια (preposition) +μένω (verb-so augment occurs before that part) -διέμενα.
Just for the record there are also some verbs that are augmented in the third singular and plural of past simple of passive voice without having an one-syllable theme, and the "augment" syllable is not accented στέλνω-εστάλη (was sent), but that should not bother you for now.
Lastly, some verbs that begin with a vowel can also be augmented and that vowel becomes η-. Ελπίζω (I hope)-Ήλπισα (i hoped) but you can choose not to augment those verbs so έλπισα is correct also (without augment those verbs sound colloquial but are grammatically correct).