There are different categories these shows fall into, all of which are very popular. There are two main categories of which many subcategories fall under, like Comedy, Romantic Comedy, Melodrama, Thriller, Horror, etc. Here they are at a glance:
Historical (Sageuk in Korean): Dramas about the Korean royal families throughout generations. These are usually fictional stories based on factual persona, but there are some that try to tell the true story behind certain events, of course some things need to be changed because some things were either lost or not recorded in their records.
Modern: Dramas set in the current time period. You will find more dramas in this category than in historical, but they are equally as polarizing. School based dramas are a big part of this category. On more than one occasion, be prepared to have your emotions checked!