Map from Google - South_Korea

The climate of the Republic of Korea (or South Korea) is cold but sunny in winter, while it’s hot and sultry with abundant rainfall in summer. Spring and autumn are quite short but pleasant. The latitude of the country is not high: the border with North Korea is located on the 38th parallel, but the winter is cold because of the prevailing winds from Siberia.
South Korea is affected by the Asian monsoon regime, so that in winter, cold air masses from the Asian continent prevail, while in summer, warm and moist air masses from South-Eastern Asia move frequently over the country. In the second part of summer and in early autumn, it can be reached by typhoons.
The precipitation pattern is the opposite to that of the Mediterranean climate: the driest period is winter, while the rainiest months are July and August.
Here is the average precipitation in Seoul.
Average precipitation - Seoul
SeoulJanFebMarAprMayJunJulAugSepOctNovDecYear
SeoulJFMAMJJASONDY
Prec. (in)0.811.833.95.31313.85.522.2153
Days767891016149797108

The total annual rainfall is significant, spanning generally from 47 to 59 inches, with the exception of the southern coast, which can reach 67 in, as well as the northern inland regions, which can go down to 40 in, and also for some particularly sheltered valleys like that of Daegu (or Taegu), which do not exceed 40 inches per year.

Winter is cold, with average temperatures below freezing in the north and in inland areas, while it’s milder, but still with night frosts along the southern coast. The sun often shines, and sometimes snow can fall, but it’s relatively rare and not abundant. The wind often blows from the continent, cold, dry and sometimes full of dust. Thanks to the greater exposure to cold winds of Siberian origin, at a given latitude the winter is colder on the west coast than on the east coast. The average January temperature exceeds 0 degrees Celsius only on the southern coast, while on the east coast it's around freezing even in the north. The interior is obviously colder than the coastal areas, in fact in the north-central inland area the average temperature in January is around 25 °F, as we can see from the temperatures of Chuncheon.
Average temperatures - Chuncheon
ChuncheonJanFebMarAprMayJunJulAugSepOctNovDec
ChuncheonJFMAMJJASOND
Min (°F)182132415261706859453423
Max (°F)323750637379828275645036

The capital Seoul is located near the coast, but in the north-west, which is the most exposed region to cold winds, and its average temperature in January is 28 °F.
Average temperatures - Seoul
SeoulJanFebMarAprMayJunJulAugSepOctNovDec
SeoulJFMAMJJASOND
Min (°F)212534455564727263503727
Max (°F)363950647381848679685439

Near Seoul, the Yellow Sea is very cold in winter, but warms up a lot in summer, reaching 75 °F in August.
Sea temperature - Incheon
IncheonJanFebMarAprMayJunJulAugSepOctNovDec
IncheonJFMAMJJASOND
Sea (°F)414141465563727573665748

In the southernmost cities like Mogpo and Busan (or Pusan), the climate is much milder, and the average temperature in January goes up to around 37.5 °F.
Average temperatures - Busan
BusanJanFebMarAprMayJunJulAugSepOctNovDec
BusanJFMAMJJASOND
Min (°F)303241505764727366574636
Max (°F)464855647275818479726150

At Busan, and along the southern coast, the sea is much less cold in winter, while in summer it gets a bit warmer than in Seoul, reaching 79 °F in August.
Sea temperature - Busan
BusanJanFebMarAprMayJunJulAugSepOctNovDec
BusanJFMAMJJASOND
Sea (°F)575555576368737975706661

The southern island of Jeju (or Cheju) is even milder, and its average temperature in January is about 41/43 °F.
Average temperatures - Jeju City
Jeju CityJanFebMarAprMayJunJulAugSepOctNovDec
Jeju CityJFMAMJJASOND
Min (°F)373943505766737568595041
Max (°F)464855647277848679706152

During cold spells, the temperature can drop to -4 °F in Seoul and the northern inland areas, and around 14 °F in the south, while it drops only a few degrees below 32 °F in Jeju-do.
The north-east is mountainous, and in addition to being freezing cold, it is also quite snowy. This is the region (called Gangwon-do) where there are more ski lifts, and given the temperature, they are also found at quite low altitudes (below 3,500 feet). The highest mountain in the area is Seorak, 5,604 feet high, protected in a national park.
The northeast coast, though milder, is a little wetter in winter: during winter the average monthly precipitation along the north-east coast is about 1.77/2.36 in, while in Seoul and in southern cities like Sokcho and Gangeung (or Kangung) it's about 0.8/1.0 in.
The least snowy area is the south-east coast, due to the higher temperature and also to the fact that in periods of bad weather the wind blows from the south. There are on average 25 days with snow (perhaps light) per year in Seoul (located in the northwest), 22 days in Incheon (near Seoul but on the coast), 27 in Mokpo (on the south-west coast), 12 in Daegu (in the interior of the south-east), and only 5 in Busan, on the southeast coast.

Konjiam, ski lift near Seoul

Spring, from March to May, is initially cool, and gradually becomes milder, but even the rains become more frequent, due to the formation of low pressure systems.
In winter and especially in spring (March to May, which is called yellow dust season) South Korea (but especially the Seoul area) can be reached by yellow dust (HwangSa in Korean), coming from the desert of Mongolia and China, which forms a characteristic haze and brings breathing problems and eye irritation.

Summer is hot, humid and rainy. The high humidity makes the weather sweltering, especially in July and August. Rainfall can be heavy especially in July and August. Two thirds of the annual rainfall occurs in summer. However, between periods of bad weather, there can also be sunny days, which may be hot.

Especially between late June and early August, an almost stationary front (called Changma) moves over Korea, which can cause heavy rainfall, associated with floods, but also fog, low clouds and drizzle. In the second part of summer, it goes a bit better and the sunshine amount increases, although there can be the contribution to rainfall given by typhoons.
In fact, from August to early October (more rarely in July), South Korea can be hit by typhoons. Typhoon is the name given to tropical cyclones in East Asia. Typhoons are accompanied with large amounts of rainfall and strong winds, and usually cause the greatest damage on the south coast.

Autumn is pleasant, especially in the month of October, in which the maximum temperatures are around 68/72 °F. Autumn is generally less rainy than spring.

When to go

Considering both the cold winter and the hot and rainy summer, the best times to visit South Korea are spring and autumn, especially the months of May and October, with a preference for October because it's less rainy (and outside the so-called yellow dust season). Moreover, in autumn you can admire the fall foliage in national parks.
As we have seen, the sea is warm enough for swimming from July to September, especially in the south-central.

What to pack

In winter: for Seoul and the north-central, warm clothes, down jacket, gloves, scarf, hat. For the south, Mogpo and Busan, warm clothes, sweater, jacket; scarf, gloves and hat for the evening, possibly a raincoat or umbrella.
In summer: light clothes, of natural fibers, light raincoat or umbrella, scarf for the wind, light sweatshirt for air-conditioned places.
In homes and temples, it is customary to remove shoes before entering: it can be useful to bring shoes without laces, or at least easy to take off.