Learn Russian: Она пьёт чай из моей любимой чашки. - She is drinking tea from my favorite cup.

The preposition из (from/out of) always requires the genitive case. Since чашка (cup) is a feminine noun, it changes to чашки. The words for my (мой) and favorite (любимая) must match its feminine genitive form, so they become моей and любимой.

I thought the genitive form of my was моего. Why is it моей here?

You use моего for masculine and neuter nouns, like my brother (моего брата). Because чашка is a feminine noun, you have to use the feminine genitive form, which is моей. Feminine adjectives also take the -ой ending in this case, which is why we see любимой.

If she is drinking the tea, shouldn't чай be in the accusative case since it is the direct object?

It actually is in the accusative case! Because чай (tea) is a masculine inanimate noun (a non-living thing), its accusative form looks exactly the same as its dictionary (nominative) form.

What is the difference between из and от? Don't they both mean from?