The form -아/어서 is the basic form to express reason and cause of something. However, it has another use, you can also use it to express a sequence between two actions where one action cannot happen without doing the other action. It is less complicated than it sounds though.
Reason And Cause
The main use of the form -아/어서 is to express the reason or the cause of what follows in the clause after it. It is the most basic form to express these things. Though there are limitations to this form, so you will have to use other more specific forms like -(으)니까, 때문에, -거든요. Though when in doubt which forms to use -아/어서 can be a good fall back option.
Also, be aware that the form -아/어서 is used idiomatically and thus its use is fixed for those expressions.
Lastly, the connective adverb 그레서 is derived from this form. Its use is the same.
가: 길이 막혀서 늦었어요.
나: 괜찮아요. 저도 금방 왔어요.
A: The traffic was jammed so I.m late.
B: It’s alright. I just arrived as well.
Another use is to express a sequence of actions. The sequence expressed is that one action happened before the other action. Furthermore, the two actions in the sentence are very closely related. In fact, the second action cannot happen if you do not perform the first action. Keep this in mind.
Furthermore, while the form is -아/어서, oftentimes with certain verbs the 서 is simply omitted, leaving simply -아/어. This is done often for verbs like 오다 (to come), 가다 (to go), 서다 (to stand), etc.
가: 엄마 배가 고파요.
나: 그럼, 과일을 한 개 씻어서 먹어.
A: Mom, I’m hungry.
B: Then wash a piece of fruit and eat it.
How To Use
The conjugation of the form -아/어서 is not too difficult. Though there are a few things to keep in mind. First, the main rule is the ‘last vowel’ rule. This means you will have to use -아서 if the last vowel in the verb stem is either ㅏ or ㅗ. In all other cases, you use -어서. The copulative verb 이다, however, is a bit special. Normally it becomes 이어서. However, in conversations -(이)라서 used. Keep this in mind. Of course, like always you will have to keep the irregular verbs in mind.
As mentioned earlier, there are a few restrictions for -아/어서. First, this form cannot utilize tense markers like -겠- or -았/었-. If it is an absolute necessity, you can use other forms like -(으)니까. Secondly, you cannot use propositive endings, like -(으)ㄹ까요, and imperative endings, like -(으)세요, in the clause after -아/어서. For this, you use forms like -(으)니까. Lastly, when you use it to express the sequence between two actions, you cannot use adjectives.