"Elder brother" or "older brother"?

They are both essentially the same and are for the most part interchangeable and no one would notice or correct you. That said, 'elder' will typically sound a little more formal than 'older'

In typical usage, I personally would say "My/his/her/their older brother" when speaking of a specific person's sibling. I would say "The elder brother/sister/sibling" to refer to the oldest of a group of siblings. I suppose that also brings out another point: I would typically use elder only when referring to the oldest of the group (I would not say "his elder brother" to refer to the the second-born from a set of three siblings, where 'his' refers to the third-born).

That said, they are almost completely interchangeable and at worst will lead to mild confusion as to which older/elder person is being referred to when there are multiple choices.