"I teach Korean."
Translation:저는 한국어를 가르칩니다.
If you still confused with the use of topic marker particles or any particles that is used here and you have an access to a library nearby, you can easily read them at "Using Korean: A guide to contemporary use" by Cambridge University Press and written by Miho Choo and Hye-Young Kwak at Chapter 19, Particles. From page 240 to 268. You will find it easy to understand, although you need to translate a little bit of Korean in examples which writer give to explain those each particle usages. I use the 2008 version. I hope it helps.
한국이 means Korea, while 한국어 means Korean language. "어" signifies that what your talking about is a language.
As for the particles:
를 is an object marking particle, which signifies that the noun that "를" is attached to will go with the verb in the sentence.
에 is a location marking particle, which means "at". It expresses the location where something "is" or "exists".
에서 is also a location marking particle, but this one expresses the location where an action is taking place, and the meaning of "from" a place.
가/이 are subject marking particles, which signify the subject of the topic of the sentence, or the subject of what's happening in the sentence. Use 이 when a word ends in an consonant, and use 가 when a word ends in a vowel. They do this to make it easier to pronounce
는/은 are topic marking particles, which signify the topic of the sentence, or what the whole sentence is about. (That's why you use 저는 in this sentence, YOU are the topic, since YOU are teaching Korean). Use 는 when the word ends in a vowel, and use 은 when the word ends in a consonant. Same thing as above, they do this for easier pronunciation.