"My daughter does not like her brother."
Translation:Mia filino ne ŝatas sian fraton.
You use si (instead of li, ŝi, or ili) when you want a pronoun referring to the subject of the verb.
In english, when you say something like, e.g. "My friend Sofia came over for dinner. My daughter does not like her brother." it might mean that your daughter doesn't like her own brother, or that maybe she doesn't like Sofia's brother. We'd rely on context to sort out whether "her" means the subject of the sentence (my daughter) or some other "her" that's relevant to the current conversation. Esperanto (and I'm pretty sure quite a few nekonstruitaj lingvoj) makes you distinguish these cases though:
Mia filino ne ŝatas sian fraton
is unambiguously "my daughter doesn't like her [my daughter's] brother"
Mia filino ne ŝatas ŝian fraton
is "my daughter doesn't like her [referring to some understood or previously mentioned person other than my daughter] brother"
Probably the most common thing you see is that si occurs in the direct object of a sentence like in the example given, but you can also have conjunctions like
Mia filino kaj sia frato ne plu malŝatas unu la alian ("My daughter and her brother don't dislike each other anymore")