Carta vs. letra
While both carta and letra can be translated as letter in English, they refer to completely different concepts in Spanish. Carta is a written message or a card, while letra is a character in the alphabet or the text of a song.
Carta
A1Carta primarily refers to a letter you write and send to someone, like mail. It also has other important meanings, such as a restaurant menu, a playing card, or a formal charter/map.
Le escribí una larga carta a mi abuela para su cumpleaños.
(I wrote a long letter to my grandmother for her birthday.)
¿Podemos ver la carta de postres, por favor?
(Can we see the dessert menu, please?)
El mago le pidió que eligiera una carta de la baraja.
(The magician asked him to choose a card from the deck.)
La Declaración Universal de Derechos Humanos es una carta fundamental.
(The Universal Declaration of Human Rights is a fundamental charter.)
El capitán del barco consultó la carta de navegación.
(The ship's captain consulted the nautical chart.)
Letra
A1Letra refers to a single character of the alphabet (A, B, C). By extension, it can mean handwriting, the lyrics of a song, a financial installment, or the literal text of a law or document.
La palabra 'sol' tiene tres letras.
(The word sun has three letters.)
Mi médico tiene una letra muy difícil de entender.
(My doctor has very difficult handwriting to understand.)
Me encanta la letra de esa canción; es muy poética.
(I love the lyrics of that song; they are very poetic.)
Este mes tenemos que pagar la letra del coche.
(This month we have to pay the car installment.)
Debes seguir las instrucciones al pie de la letra.
(You must follow the instructions to the letter.)
Summary
In short, use carta for a physical letter (mail), a menu, or a playing card. Use letra for a character of the alphabet, handwriting, or song lyrics. Carta is a complete message or object, whereas letra is a fundamental component of writing or music.