Quoted from

http://www.howtostudykorean.com/unit1/unit-1-lessons-1-8/unit-1-lesson-5/

Notice that some of these grammatical principles require the addition of “~아/어.” Many grammatical principles (or conjugations, or any other thing) require the addition of “~아/어” to the stem of a verb or adjective. Notice that the “/” indicates that you need to choose what actually gets added to the stem. In some cases it is “~아”, and in some cases it is “~어”. The following is the rule that you can use to determine if you should add “~아” or “~어”:

If the last vowel in a stem is ㅏ or ㅗ (except 하) you add ~아 followed by the remainder of the grammatical principle
If the last vowel in a stem is anything but ㅏ or ㅗ you add ~어 followed bythe remainder of the grammatical principle
When conjugating to the past tense, we need to add “~았/었다” to the stem of a word. Following the rule above, ~았다 is added to words with the last vowel being ㅗ or ㅏ and ~었다 is added to words with the last vowel being anything but ㅏ or ㅗ. For example:

나는 먹다 = I eat (note that this sentence is unconjugated)
The last vowel in the stem is ㅓ. This is not ㅏ or ㅗ. So, we add 었다 to the stem:
나는 먹었다 = I ate (먹 + 었다)

나 문을 닫다 = I close the door (note that this sentence is unconjugated)
The last vowel in the stem is ㅏ. So we add 았다 to the stem:
나는 문을 닫았다 = I closed the door (닫 + 았다)

나는 창문을 열다 = I open the window (note that this sentence is unconjugated)
The last vowel in the stem is ㅕ. This is not ㅏ or ㅗ. So we add 었다 to the stem:
나는 창문을 열었다 = I opened the window (열 + 었다)

What makes this complicated (at first) is that for verbs that have a last syllable that end in a vowel, the 았다/었다 gets merged to the actual stem itself. This is how 아 and 어 merge with syllables ending in a vowel:

아 + 아 = 아 (example: 가 + 았다 = 갔다)
오 + 아 = 와 (example: 오+ 았다 = 왔다)
우 + 어 = 워 (example: 배우+ 었다 = 배웠다)
이 + 어 = 여 (example: 끼+ 었다 = 꼈다)
어 + 어 = 어 (example: 나서 + 었다 = 나섰다)
여 + 어 = 여 (example: 켜다 = 켰다)
** When the last syllable of a word is 하, it gets conjugated irregularly (thousands of words end with the stem 하). Instead of adding 아 or 어 to the stem, you add 여 to word stems that end in 하. (하+여 = 하여). 하여 is usually written/spoken as “해”, although there will be some situations (usually official documents) where you will see “하여” used instead of “해”:
하 + 여 = 해 (example: 이해하 + 였다 = 이해하였다 = 이해했다)
For words where the last vowel is ㅡ, it is very complicated and will be covered in the next lesson.
Many people have asked me “what if the last vowel in a stem is a more complicated vowel, like ㅠ, ㅑ, ㅔ, etc…?” You will find that the stem of almost all verbs and adjectives in Korean do not end in these complex vowels. The most common words I can think of that have stem that ends in one of these complex vowels are:바래다 (to fade)
매다 (to tie up)
메다 (to put on/carry something on one’s shoulder)With these words (and others like it), the same rule applies as above. That is, the final vowel does not end in ㅏ or ㅗ, so we need to add “어” plus whatever we are adding. With these complex vowels, it is irrelevant if you merge the addition to the stem. Both forms (merged and non-merged) would be correct. For example:바래 + 었다 = 바랬다 or 바래었다
매다 + 었다 = 맸다 or 매었다
메다 + 었다 = 멨다 or 메었다
Here is a more detailed breakdown:

가다 = to go
The last vowel in the stem is ㅏ. So we add 았다 to the stem.
나는 가았다
But, because the stem ends in a vowel, 았다 can merge with 가:
나는 갔다 = I went

오다 = to come
The last vowel in the stem is ㅗ. So we add 았다 to the stem.
나는 오았다
But, because the stem ends in a vowel, 았다 can merge with 오:
나는 왔다 = I came

배우다 = to learn
The last vowel in the stem is ㅜ. So we add 었다 to the stem.
나는 배우었다
But, because the stem ends in a vowel, 었다 can merge with 우:
나는 배웠다 = I learned

던지다 = to throw
The last vowel in the stem is ㅣ. So we add 었다 to the stem.
나는 던지었다
But, because the stem ends in a vowel, 었다 can merge with 이:
나는 던졌다 = I threw

건너다 = to cross
The last vowel in the stem is ㅓ. So we add 었다 to the stem.
나는 길을 건너었다
But, because the stem ends in a vowel, 었다 can merge with 어:
나는 길을 건넜다 = I crossed the street

Examples:
나는 친구를 만났다 = I met friends
나는 밥을 먹었다 = I ate rice
나는 한국어를 공부했다 = I studied Korean
나는 공을 던졌다 = I threw the ball
나는 길을 건넜다 = I crossed the street