Carta means card (card games) and letter, but in this case it's letter because Duolinguo uses the word Letter in the beginning exercices, i noticed that studing french and spanish. It's the Duolingo pattern that is abit unflexible to portuguese language. Other thing is one word in portuguese CAN MEAN 1 TO MANY THINGS IN ENGLISH, that is because english has a better expressions variety than portuguese (in vocabulary case). But what portuguese best has to offer is the tricky grammar that if explored wisely can be very interresting and flexible. It's a matter of time so you guys start to get delighted with brazilian literature.
Well...you might have to wait for a native speaker to respond with a better explanation, but I've always kept it in my mind that "carta" is more for letters and other things that you write on (Like greeting cards), while "card" (like a business card or credit card) would be the word "cartão". Anyway, when it listed card it may have been trying to hint at the greeting type of card, which somewhat falls into the "letter" category...or I could be just talking nonsense, in which case you should probably try to report the error next time. :D
Yep, letter means carta, ou letra. (A primeira letra do meu nome é P) the word for cartão / carteira is card, as cartão de crédito, carteira de estudante, cartão de memória, and so on. Those cards we give on special occasions (birthday, christmas,...) you use cartão. (Cartão de Natal, example). As a simple rule, cartões are those small square papers sometimes already written on, while carta you have to write everything by your hand, or computer.... but you have to write, or something longer, carta de admissão (admissions form). Anymay, cloudhorizon's answer is clearer than mine :)
"carta" can be be a document (a letter, a map like in "carta de navegação", or some other kinds of documents), a piece of playing cards, or a menu (in a restaurant).
"cartão" is a card (like a credit card), but can also be a kind of letter (the kind you buy already written, usually for a special date like one's birthday).
Ok, lemme try to help you out with definitions and whereabouts. "That is my letter" in clear and sound portuguese would be "Aquela é minha carta". That (Away from speaker and listener) = aquele (masculine)/ aquela (feminine) / aquilo (to a thing. If used to people than you are insulting them, take care.). Now for the next question: "It is my letter" literally means "Ela é minha carta", in portuguese you can omit or "vanish with" the pronoum Ela, so it would be "É minha carta"........ Still not sound like "A carta é minha" what is more clear and sound.