We had been exploring Gyeongju all afternoon and had started to get hungry. It looked like it was going to be a lovely sunset so we decided to get a quick meal and come back out to watch. So we walked straght into the first restaurant we found, and sat down. No menu was offered, and within a minute a huge array of small dishes began arrving at our table, unloaded from several very large trays. We were, as you might expect, a bit bemused by this and couldn't begin to fathom where we might start. We fished out the Lonely Planet and think what we'd been presented with was 'Hanjeogsik' - "a banquet that includes fish, meat, soup, tofu stew, rice, noodles, steamed egg, shellfish and lots of cold vegetables amongst many other items". It was entertaining to pick through and discover what was what, but with over thirty dishes to go at we were stuffed well before the end....
said...
Actually, almost all Korean meals are eaten like this, as you might have discovered. The oily vegetables and small meats or other marinated foods in little bowls are side dishes, and hot or boiling liquids in large pots are shared stews, and small bowls of soup are individual soups. Meat or other luxurious dishes are usually eaten at dinner or lunch. Fish is for breakfast or for a lighter meal. There are many other kinds of foods, but these are the basics for a typical Korean-style meal.
Other kinds of meals (at a restaurant) are when you grill meat on the table. This is kind of expensive, but very popular, especially among colleagues and in groups. The traditional Han-sik is with all the side dishes, your own soup, and rice. These come free, like you observed above, but you have to order some kind of meat or jjigae or something like that, and that will be what you pay for. Sometimes you can just pay for your rice, especially near a university or where there are a lot of young people.
The other meals are noodle meals, which are often really cheap and tasty. There are Chinese noodles, Japanese udon, and of course all kinds of Korean noodles and mostly noodles in a soup. Porridge is also popular, vendor food is popular, and more. Honestly, now that I've started explaining all this, I realize explaining all the kinds of Korean food-serving formats would take a book to properly explain, so I suggest if you are ever in Korea again, spend the whole time looking at foods and the excellent architecture and civic(?) design. There are other things to see, but these in my opinion the two best parts of modern Korea.