It depends on the last vowel sound of verb or adjective stem -았- is used after the vowel ‘ㅏ’, ‘ㅑ’, ‘ㅗ’, ‘ㅘ’ -었- is used after the vowel ‘ㅓ’, ‘ㅕ’, ‘ㅜ’, ‘ㅡ’, ‘ㅣ’, ‘ㅐ’, ‘ㅔ’, ‘ㅙ’, ‘ㅚ’, ‘ㅟ’, ‘ㅢ’ -였- is used after the verb or adjective of '하다' ex) 가다(go) verb stem is 가 The last vowel of 가 is ㅏ 가 + 았습니다 => 가았습니다(omitted duplicated ㅏ) => 갔습니다 울다(cry) verb stem is 울 The last vowel of 울 is ㅜ 울 + 었습니다 => 울었습니다. 잡다(take) verb stem is 잡 The last vowel of 잡 is ㅏ 잡 + 았습니다 => 잡았습니다 오다(come) verb stem is 오 The last vowel of 오 is ㅗ 오 + 았습니다 => 오았습니다(contracted ㅗ and ㅏ to ㅘ) => 왔습니다 공부하다(study) verb stem is 공부하 공부하 + 였습니다 => 공부하였습니다(contracted ㅏ and ㅕ to ㅐ) => 공부했습니다 There are some irregular verb or adjective 슬프다(sad) ㅡ irregular verb stem is 슬프 The last vowel of 슬프 is ㅡ 슬프 + 었습니다 => 슬프었습니다(omitted ㅡ ) => 슬펐습니다 돕다(help) ㅂ irregular verb stem is 돕 The last vowel of 돕 is ㅗ 돕 + 았습니다 => 돕았습니다(changed ㅂ to ㅗ ) => 도왔습니다 부르다(call) 르 irregular verb stem is 부르 The last vowel of 부르 is ㅡ 부르 + 었습니다 => 부르었습니다(omitted ㅡ and doubled ㄹ ) => 불렀습니다

well, it is a form of past tense in korea. we use ~았습니다 when the vowel ending is ㅏ or ㅗ. and 었습니다 is used when the vowel is 이 으 우 and we use 했습니다 when the ending using 하다