"The woman studies Korean."
Translation:여자가 한국어를 공부합니다.
The order of the subject and object dont actually matter (other than emphasis) but you will get it wrong if you swap words around, apparently.
공부하다 is to study and 배우다 is learn. Studying is working to commit information to memory, learning is assimilating new information into one's mind to apply it in different contexts. You can study physics all night and then fail your exam the next day, so you didn't learn anything. Likewise, you can learn to speak English with complete strangers by watching the TV show Friends, which isn't really studying it.
I am not sure but I feel like it is better not use 를 here because technicaly it could've been 여자가 한국어공부를 합니다or something like that, but then again Im not korean so please check it one more time just in case
공부 is a noun derived from a Chinese word; 하다 "to do" is used to turn Sino-Korean nouns like this into verbs. It's also used in this way for words similarly borrowed from other languages. Think of it like adding "-ing" to the end of a noun in English to make it a verb.