I've read both forms in newspapers and online news: elder brother and older brother. What's the difference between them? When should I use which?
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.
Basically, they are the same, but there is a slight difference in their meaning. When referring to someone who is older than you in your family or someone who has a close relationship with you, you should use "elder".
The usual comparative and superlative forms of old are older and oldest: •My brother is older than me. •The palace is the oldest building in the city. In literary or formal writing, elder and eldest may be used when comparing the ages of people, especially members of the same family. As adjectives, they are only used before a noun and you cannot say “elder than”: •my older/elder sister •the older/elder of their two children •I’m the oldest/eldest in the family.
Oxford Advanced American Dictionary s.v. elder