Japan has apologized to Korea
The Japanese have repeatedly attacked Korea over the last few centuries. As a result of this, Korean have an understanably hostility towards Japan. A few examples of Japan attacking Korea are:
From the Silla Dynasty (668-935 AD) to the ending states of the Goryeo Dynasty (935-1392), Japanese pirates were a problem in Korea. Near the end of the Goryeo Dynasty, Japan began to be more aggressive; actively sending military campains against Korea. General Yi Seonggye (later known as King Taejo) was largely responsible for driving the Japanese back. He was also the man to usher in the Joseon Dynasty (1392-1907).
Throughout the Joseon period, the Japanese invaded Korea many times. The most famous of these was throughout the 7-Year War (1592-1598), when Toyotomi Hideyoshi (many liken him to Hitler in many respects) was emperor of Japan. There are numerous stories of the Japanese brutally maiming Koreans during this time. General Yi Sun Shin (a hero for many Koreans. He was a really cool character, I must admit) was one of the main people, as well as allies in Ming Dynasty China, responsible for repelling the Japanese back to Nihongo with what many historians give the nod to as the first ironclad ship, called a 'turtle ship.'
The Joseon period ended in the early 20th century, when Japan created a protectorate (1907) over, and later annexed (1910), Korea. This lasted until the end of the Japanese Empire, following Japan surrendering in World War II (1945). There were countless atrocities committed by the Japanese during this time. You can get a small picture of this by reading my post, and looking at the pictures, of Seodaemun prison in Seoul. My heart feels heavy everytime I think about that place. You could also do an internet search on the rape/massacre of Nanking, to get an idea of what the Japanese did to Chinese people during this same time.
Yet, these events are all in the past. Although history should never be forgotten, those responsible for the sad events of the past are just that; past. Over and over again, I've heard from Korean people that Koreas would be more inclined to, well, not hate Japan so much if Japan would only apologize for it's actions against Korea. Well, that has happened a couple times that I can think of. The first instance that I came across was when I was in Hiroshima.
This statue, located in the Peace Park in Hiroshima, is a memorial dedicated to all of the Koreans that were killed by the atomic bomb in Hiroshima. I don't know if the word apologize was specifically on there, but there was definetely a feeling of regret for the Koreans that were used as slave labourers during WWII.
The second time that I've came across an apology (this one was much more direct, although it covered a broad range of time) I came across at Wikipedia.org. The current Emperor of Japan, Emperor Akihito (the son of Hitohiro, of Emperor during World War II), acknowledged his countries actions against Korea; calling them "regretable." It reads that, "On December 23, 2001, during his annual birthday meeting with reporters, the emperor, in response to a reporter's question, remarked that he felt a "certain kinship with Korea", and went on to explain his feeling as resulting from the fact that the mother of Emperor Kammu (736–806) was a descendant of Korean dynasty King Muryeong of Baekje as documented in Shoku Nihongi. The emperor also noted that Koreans who migrated to Japan in ancient times introduced important aspects of culture and technology to the country, and then called upon his countrymen never to forget the regrettable fact that Japan’s exchanges with Korea have not all been so friendly." On top of this, Emperor Aihito, as well as his father, Emperor Hirohito (since learning who was there), have boycotted visiting Yasukuni Shrine, where a number of Japanese war criminals are enshrined there (the Japanese Prime Ministers (ie. PM Koizumi) have not boycotted visiting this shrine, much to the digust of Japan's Asian neighbors, however).
I'm not exactly sure what many Koreans are expecting. Do they want Japan to bow down to Korea? I'm not sure if they would fully accept any apology from Japan. Personally, I think Koreans should work to have a more friendly attitude towards Japan. In turn, I do think that Japan should be a little more sensitive to Korea in the way they do things. It's a sad situation, especially from the view of an outsider looking in.