Yes that is the correct website.
Just start an application, you will soon get to the page where you find out that documents they want - depending on what type of visa you are applying for they require different things. One thing they will ask for is proof of your legal residency in Korea, if you are just visiting your passport and entry stamps might be sufficient although the website suggests you may need more..."For non-Korean nationals: a copy of Alien Registration Card front-back"
The website deals with applications globally but each country's Embassy might ask for more information from you and of course can refuse to issue a visa if they are so inclined.
You should print it, makes life a lot easier at check-in and immigration.