- Меня is the accusative case, used for direct objects.
- Мне is dative case, used for indirect objects.
The exact meaning depends on the verb, but generally direct object is more strongly affected by the action than the indirect object. With «дай», the thing given (я́блоко) is the direct object, and the receiver (мне) is the indirect object.
All of them are used in Russian (although the second one sounds a bit less natural than the 1st and the 3rd).