Tämä on Korea = This is Korea (the country).
Tämä on koreaa = This is Korean (the language). Note the lower case.
The name of the language is also Korea, but when you say something is in a particular language, you use the partitive case. So Tämä on suomea (from the basic suomi), Tämä on ruotsia (from the basic ruotsi), Tämä on englantia (from the basic englanti), Tämä on koreaa.
Sorry, but I still don't understand why on this case we need to use partitive. I do understand that if I'm saying that something is in korean I need to use partitive case, but in this sentence we are not talking about anything particular. It could be use by someone standing in front of Korean group of people and say "Tämä on korea", meaning that this language is korean. Am I wrong?
You are talking about something in particular, though, because you are using "tämä" to refer to it. In your example of the group of people, you need the -a.
I don't think it's helpful to always try to understand why a language uses a specific structure, it's sometimes better to just learn it. Each language has its own idiosyncrasies, and the many uses of the partitive is one of the Finnish language's. We are maybe sometimes trying too hard in these discussions to provide explanations for "why" we use the partitive, but I'm starting to notice (in other threads) that it leads to strange overgeneralizations and misunderstandings.