Learn Spanish (Latin America): Ella escribe la carta otra vez.

You would say Ella vuelve a escribir la carta. Both that and Ella escribe la carta otra vez mean exactly the same thing. Spanish speakers use both structures interchangeably to say that someone is doing an action again!

What does otra vez literally translate to?

It literally translates to another time. Otra means other or another, and vez means time (as in a specific instance or occurrence, like doing something three times, rather than the time on a clock).

Why isn't it una otra vez since in English we say AN-other?

In Spanish, the word otro (and its feminine form otra) already includes the meaning of a or an. You must never put un or una in front of it. So to say another time, you just say otra vez, never una otra vez.

Could I use de nuevo instead of otra vez at the end of this sentence?

Yes! You can absolutely say Ella escribe la carta de nuevo. Both otra vez and de nuevo are very common ways to say again in Spanish, and they can usually be used in the exact same spots in a sentence.