Busan (also named as Pusan) is the second largest city of South Korea. Its fame is due to numerous festivals, exhibitions, and events. For this reason, most of its visitors go there for a specified purpose such as the International Film Festival, not general touristic travel though Busan's Haeundae Beach is a popular summer seaside resort for Koreans.
Busan is not far away from Seoul and can be easily reached by the excellent KTX rail system in a little over 2 hours and home to the International Ferry Terminal serving Fukuoka, Shimonoseki, and even Osaka.
The city is a really nice place to be. You have everything you would expect from a large, modern city, but without the stressed-out feel that Seoul can have. In the summer you have the beaches and the aquarium, known for its "swimming with sharks" experience. In winter, there are some ice-skating rinks and some great shopping. All year round there are lots of really striking temples, each worth seeing for their own reasons, and of course, there are some great hiking trails and spas to be discovered. This is a great place to see urban life in Korea as there are much fewer tourists. Although there are some English-speaking persons there, it's much less than the countries capital.