South Korea covers the southern half of the Korean Peninsula, has a largely mountainous interior, and also includes thousands of mostly small islands off its southern and western coast. Most mountains are found in the east and southwest of the country, but South Korea's highest mountain is the extinct volcano of Hallasan (1950 m) on the the southern island of Jeju.
The main granite mountain ridge running uninterrupted from north to south along the entire length of the Korean Peninsula is called Baekdu-Daegan, or White Head Great Ridge. It starts at North Korea's highest peak, the volcanic Mount Baekdu (2744 m), on the Chinese border in the north. It ends in South Korea at Jirisan. The ridge is the watershed of the Korean Peninsula. Numerous subsidiary ranges extend from the main ridge to form a landscape of mountains, waterfalls and river valleys. Roger Shepherd from New Zealand initiated an interesting project called Baekdu-Daegan Trail, meant to one day cover the entire length (1700 km) of the ridge through both North and South Korea. At this moment the complete 735 km South Korean section is actually hikable, and documented in a guidebook. For many South Koreans the trail appears to be something of a nationalist pilgrimage.
Over the past decades hiking and mountaineering have become an increasingly popular passtime among the Koreans, especially during weekends. Hiking trails have been laid out in all of Korea's National Parks, and in other places local initiatives have resulted in a growing number of trails. Most popular National Parks for hiking are Songnisan, Seoraksan, Jirisan and Wolchusan National Parks. Most parks have campgrounds and sometimes cabins to stay overnight. If you travel to Jeju Island, the Hallasan National Park offers wonderful day hikes through a volcanic landscape of craters, lakes and plateaus. To get an impression of Korean national parks and their hiking trails, take a look at the excellent website of the Korea National Park Service.
The best season to go hiking in South Korea is in spring (March till May) and autumn (September till early November). Summer can get very hot and humid, and in winter it is freezing cold with most mountains covered in snow.