"Give me an apple, please."
Translation:Дайте мне яблоко, пожалуйста.
Я is used for the subject of the sentence. Someone who is doing the action. So, «Я даю» means 'I’m giving, I give’. This form is called nominative case.
Мне is used for an indirect object, someone who is affected by the action, but not directly. In «Дайте мне яблоко» 'Give me an apple', яблоко is a direct object, because the ‘giving’ affects apple the most (it’s being moved around), and мне is the indirect object, becuase I am affected by ‘giving’ (I get one apple), but not to such an extent as apple is. Мне is the dative case form.
Russian has more cases, and you’ll learn them and their usage as you progress through the course.
By using the perfective verb conjugated in present it really means a future. While conjugating in the imperfective form would mean a present action in process. So we use here the perfective form because you want him (or her) to give you the apple, he is not giving you the apple already, so the this is a future.