However, students aren't free to relax, go on vacation, or even really get much of a break during these "vacation" periods because they are required to attend "camps". Camps (and for us native English teachers, English camps, specifically) are additional special courses that students must enroll in during their summer and winter vacation times.
This is similar to how kindergarten started out as completely optional in the USA--wait. Did you even know kindergarten was optional in the USA? In fact, even today only 16 states mandate kindergarten enrollment! Today, kindergarten is seen as such a norm and necessity we don't even question the idea of enrolling our children in it. Parents don't want their children to be "left behind" their classmates academically and socially upon entering 1st grade, so the idea not enrolling them isn't even given a second thought.
This is essentially the mentality behind Korean summer and winter courses, as well as the hagwon (학원: Korean private school/academy) obsession pushed upon Korean students today. Parents, and many times even the students themselves, are so obsessed with getting that extra leg-up against their peers that they're willing to sacrifice nearly any and all free time. Of course (just like kindergarten enrollment in the States), when everyone does it, it eventually becomes expected. To not attend winter or summer camp would be seen as quite strange or lazy.
Most native English teachers (NETs) are asked to teach two weeks of camp in the winter, and two in the summer. Some schools condense it into one, and others have even extended it into three or four. It all depends on the needs of the school and the teacher's mandated vacation time, which is why teachers at different schools will have different breaks and camp requirements. As contracted public school teachers in Jeollanamdo, our contracts entitle us to 24 paid vacation days in winter and 8 in summer. This means that for the long winter break of January and February, any days that are not in my 24 approved days I must be in school as I'm being paid for them. Most schools will have teachers "desk warm" during that time (no classes to teach, just required to be present in the office), though the school has the right to utilize us for camp classes during those days if they so desire.