The word "se" is used whenever you want to point that something is part of, property of, like when you use "for himself/herself" or "his/her".
So putting this as examples:
Mi hermano menor se rompió la mayoría de vasos.
means "my little brother broke the mayority of his glasses" (but since the bother doesn't have glasses inside his body nor the glasses are part of the body, then it makes no sense).
And when you say:
Mi hermano menor rompió la mayoría de vasos.
it means "My little brother broke the majority of the glasses"
Now adding a bit more understanding example denoting property:
When you want to say "My brother bought a book for himself" (clearly states that the book was bought for him and not for someone else)
here are the two comparatives:
"Mi hermano se compró un libro"
here you are clearly stating that the book was bought for him.
"Mi hermano compró un libro"
here you are saying that your brother bought the book but this doesn't explain for who.
Now adding a bit more understanding example denoting that something is "part of":
When you want to say "My brother broke his arm" (clearly stating that the arm broken is his) here are the two comparatives:
"Mi hermano se rompió el brazo"
here it's clearly that the arm broken is his and not someone else's
"Mi hermano rompió el brazo"
here you are just saying that he broke an arm but it doesn't mean that the arm broken is his, since he could have break someone's else.
I hope it helped to make it a bit more clear.
Now clarifying your doubts for this sentence:
A mi hermano se le rompieron la mayoría de los vasos.
Why "le" is used? What does it refer to? Is it "pronombre acusativo" or "pronombre dative"?
"le", "lo" , "los" and such are accusative pronouns. "le" is used to denote that the glasses were broken by him.
What kind of sentence structure is it? I wonder why when we say "a mi hermano...le...", it means the "hermano" breaks the glasses by
because the glasses could have been broken by something else (maybe a strong wind blew the glasses away), its pretty much like when you say: "my brother cut his finger" and "the finger of my brother was cut"; in the first one the finger was cut intentionally and in the second one the finger could have been cut by accident (i.e. a paper sheet)
What is the subject and object for this sentence?
The subject is "mi hermano" (my brother) and the object "los vasos" (the glasses).
Is "se le rompieron" reflexive? Who do the action on the glasses? Who receives the action?
No its not reflexive since he is not breaking his own glasses (since a person does not have glasses inside his body or as part of his body). Reflexive verbs show what a person does to him/herself.
"mi hermano" (my brother) does the action on the glasses, "los vasos" receives the action since they are beeing broken by my brother.
Here is a great slide about Reflexive Verbs