"Eat the apple!"
It says the correct translation is "elmayi (-dot) yiyin", why not "elmayi yiyorsun"? And "elmayi ye!" is also correct?
The second way?
There is no second way. It's just that if you're talking to more than one person (or talking to one person with respect [the Dutch U]), you add "-(y)in" to the "regular" imperative.
Yap! → Yapın!
Söyle! → Söyleyin!
"Ye" is just an exception; it's not "yeyin" but "yiyin!".
The verb "Demek" (to tell), for instance, which always follows the same format as "yemek" is regular in this case: we write "Deyin!" (although it's pronounced 'diyin').