"Give me juice, please."
Translation:Дай, пожалуйста, сок.
Мне is the dative case of я. Dative case in Russian is used for indirect objects, such as «дайте мне сок», "give to me juice."
Direct objects are the objects directly affected by the verb (the juice is the thing being given, not you).
Indirect objects are the objects receiving the direct objects (you are the one receiving the juice).
For another example that doesn't use "give": in the sentence "please tell your brother a joke", the joke is the thing being told, so it's the direct object, and your brother is the object receiving the joke, so it's the indirect object.
In both of these examples you can rearrange the sentence to see that the indirect object is receiving. "Give juice to me"; "Tell a joke to your brother".
It's worth noting that this is not true in a sentence like "I am going to the store." In this case, the store is the object of the preposition "to". You can tell because if you change the preposition to something like "near the store" it still works just fine. In this sentence you would use the prepositional case for "store."