Learn Russian: Она изучает немецкий язык.

The verb изучать (here as изучает) is used when you are studying a specific subject or discipline, like a language, math, or history. It always takes a direct object. In contrast, учиться means "to be a student" or "to study at a place" (like at a university) and doesn't take a direct object.

Is there a difference between "studies" and "is studying" in Russian?

No, Russian doesn't have a separate "continuous" tense like English does. The single present-tense verb изучает translates to both "she studies" (in general) and "she is studying" (right now), depending on the context.

Why do we need the word язык (language)? Can't we just say Она изучает немецкий?

In Russian, when you talk about studying a language, you generally pair the adjective немецкий (German) with the noun язык (language). While people sometimes drop it in very casual, everyday speech, it is grammatically expected to include язык. Note: If you want to say "She speaks German," you use a completely different word, the adverb по-немецки.

What case is немецкий язык in?

It is in the accusative case because it's the direct object of the sentence (the thing being studied). However, because язык is a masculine, inanimate noun (it's an object, not a living person or animal), its accusative form looks exactly the same as its basic dictionary (nominative) form. That's why the ending doesn't change here!