Basically Korean uses the same system of Western society (Gregorian calendar) to represent dates and times. The standard time zone of South Korea is GMT+09:00 without the summer time period. It's same to Tokyo, Japan.

For traditional holidays, we count them on a lunar calendar which is based on Chinese calendar. We don't have any traditional holidays based on weekdays like United States (eg. Thanksgiving day is the fourth Thursday of November in U.S.) because we didn't have the weekday concept before Western culture was introduced.

We had a very different calendar system at the past, which divided a year into 24 periods and count the year based on the starting years of kings' ruling period¹ or 60 years of "Kanji" cycles. Before using the concept of weeks, China and Korea usually divided a month into 5 or 10 days. I will introduce this traditional calendar system on another post later.

Weekdays

You may notice the name of monday and sunday have the same meaning to English ones. Also other names have some relations to English names.³ For example, saturday is come from 'Saturn' which also means the sixth planet of our solar system and we say that planet as '토성(土星)'. These weekdays names are influenced by Japan that uses same names.⁴

5-days work on a week is now widely accepted in South Korea, so the friday is beginning of weekends like most Western countries, but without shortened work time at that day. (In Sweden, the work time on fridays is often shorter than other weekdays, but in Korea, there is almost no exceptions.)

Saying Years, Months and Days

We write dates in 'Year Month Day' order while Americans use 'Month Day Year' and Sweden uses 'Day Month Year'. Almost everybody knows English month names though they are not used frequently in Korea, and there are no month names in (modern) Korean.⁵

So we just say January as '1월', February as '2월', ..., December as '12월'. The character '월' is Korean pronunciation of Chinese character '月' which means month or the moon. All numbers in dates are said in the cardinal number form.

Generally, year is '년(年)', month is '월(月)', and day is '일(日)' or '날'⁶.
Example: April 12, 2008 = 2008년 4월 12일 (이천팔년 사월 십이일)

Usually 3000BC is said '기원전(紀元前) 3000년' and AD2000 (or CE2000) is said '기원후(紀元後) 2000년' in some historical contexts. '기원' means a specific moment of a historic event depending on the context--here, the birth of Jesus Christ.

Saying Time

To say hours in Korean, you should use ordinal numbers, but cardinal numbers for minutes and seconds.

I think it would be faster to look some examples.

We don't have some varied expressions to say times such as 'ten past twelve', 'a quarter past three', etc in English, but just say the exact numbers.

General Nouns about Times

Time Span

We use '시간(時間)' instead of '시(時)' and other parts are same.⁷