I grew up in India and am familiar with this, used to always wonder if I was wrong and why i often get confused when people say parentheses instead of brackets.(I have been in the US since the last 10 years)

Interesting, I'm from the US and I do call () parenthesis but the last two are the same from what I've learned and what other people say (usually in math classes).

In New Zealand:

  () == "round brackets"
  [] == "square brackets"
  {} == "curly brackets"
  <> == "pointy brackets"

In England I learned the version further up this thread. Computer Science nomenclature doesn't really follow British vs. American English separations as much as natural language.

In fact I don't recall anyone in Britain use those terms to me, either in university or at work.

That's how I learned it (although I never used 'round brackets' explicitly). This gets really confusing when you begin to talk to Americans and refer to 'brackets', thankfully everybody understands parens, and the others are the same.