Some examples:
Native Korean
- Counting physical objects (including people) and usually followed by a countword (개, 명, 분, 잔, etc. depending on what is being counted)
- Years of age
- Hours
- Counting (in general)
- Counting months (but only when used with the native Korean word for month/moon: 달)
Sino-Korean
- For any number 100 or higher even if it would normally be expressed in native Korean (since native Korean only goes to 99)
- Counting any unit of time other than hours (if counting months, you follow it with 개월, which is the generic countword + the Sino-Korean word for month/moon)
- For creating the names of the months (number of month + 월)
- Money / currency
- Math
- Measurements (kilometers, grams, liters, etc.)
- Phone numbers
Those are all the examples, I can think of off hand.
Notice that hours and minutes do *not* use the same counting system, which means *both* systems are used when telling the time. 1:01 = 한 시 일 분 (literally: 1 hour 1 minute) and would usually be written as 1시 1분.
However, 한 should come after 감자. Sorry I can't be more elaborate with my Korean examples. I should figure out the Korean keyboards.
To elaborate on this. "1 potato" would be expressed as: 감자 한개
(개 is the generic countword for any object that doesn't have its own specific countword. However, if you forget a specific countword, 개 is usually considered safe to fall back on for anything *except* counting people.)