This was one of our definition questions. How are things like dictionaries sorted, and what is equivalent to our A-Z or Z-A lists?
Dictionary order is defined in 한글맞춤법 2장 4항
- 자음(19자) ㄱ ㄲ ㄴ ㄷ ㄸ ㄹ ㅁ ㅂ ㅃ ㅅ ㅆ ㅇ ㅈ ㅉ ㅊ ㅋ ㅌ ㅍ ㅎ
- 모음(21자) ㅏ ㅐ ㅑ ㅒ ㅓ ㅔ ㅕ ㅖ ㅗ ㅘ ㅙ ㅚ ㅛ ㅜ ㅝ ㅞ ㅟ ㅠ ㅡ ㅢ ㅣ
- 받침(27자) ㄱ ㄲ ㄳ ㄴ ㄵ ㄶ ㄷ ㄹ ㄺ ㄻ ㄼ ㄽ ㄾ ㄿ ㅀ ㅁ ㅂ ㅄ ㅅ ㅆ ㅇ ㅈ ㅊ ㅋ ㅌ ㅍ ㅎ
A typical way of memorizing these is to say '가나다라마바사아자차카타파하', '아야어여오요우유으이', then insert the appropriate double alphabets in between. As for the 받침, the order is more or less random, and only for consistency across dictionaries. I don't think a normal Korean speaker would know it (I had to look it up too).
As @Leftium pointed out, this order is the one used in South Korea. For historical and North Korean ordering see https://en.wikipedia.org/wiki/Hangul#Sorting_order.