Despite attempts to market the island as "the Hawaii of Korea," climatologically and geographically it bears little in similarity to the Hawaiian Islands in the U.S. The island offers visitors a wide range of activities: hiking on Halla-san (South Korea's highest peak) or Olle-Gil(routes), catching sunrises and sunsets over the ocean, riding horses, touring all the locales from a favorite television K-drama, or just lying around on the sandy beaches.
Although tourism is one of the main industries on the island, many of the hotels and other tourist areas are run by mainland companies, so much of the income never gets put back into the local economy. Also, since the attractions are geared toward tourists, many of the entrance fees can be hefty (although the locally owned and operated ones tend to be cheaper). Similar to Gyeongju and some other areas, local residents can enter most places for free or for a steeply discounted price.
Jeju is Korea's capital for unusual theme parks and niche commercial attractions. Oddly somewhat of a duopoly has developed on the island with typically two competing variants on the one theme in Jeju-si and Seogwipo-si, respectively. This trend extends even to the most bizarre ideas with pairs of sex, glass, and teddy bear museums all in existence.
The local traditional culture stands in stark contrast to the mainland (and much of Asia) as being matriarchal. Stemming from this basis, and some odd tax reasons, the role of seafood gathering on the island has been dominated by women. As such, a common sight around Jeju's coastline is that of the "haenyo" or "woman diver", a figure that has become somewhat iconic of the island. Perhaps even more iconic are the "Dol Hareubang" or "grandfather statues", giant basalt statues. You will often see them outside restaurants, plus a few are thrown into any tourism site for good measure.
Much of the northern coast of the island is densely built up, and not particularly attractive. Some of the more attractive beaches of the island, however, are at the western part of the north coast at Hallim. The south coast of the island has more been developed as a tourist destination. Although the island is no longer than 70 km, travel times between different parts of the island are relatively high (especially by bus), and your stay on the island will strongly be affected by where you decide to stay and what transport you will be using. Renting a car may turn out to be useful.