In the UK, 'parentheses' and 'brackets' are the same: ( )
In the USA they're either 'parentheses' or 'round brackets'.
Here in the UK, I'd always call them simply 'brackets' except that I might refer to something as being 'in parentheses'.
Braces is a solely American term for what most Brits would call 'curly brackets': { }
Apart from when studying mathematics (where different types of bracket have very different meanings), almost the only time that I've ever found the need to use anything other than 'normal' brackets is when one pair of brackets is embedded between another pair. Then using curly or square brackets for one of the two pairs can make it easier for the reader to follow what is going on.
Note that in the USA, a 'bracket' is usually what Brits would call a 'square bracket'.
Confused? Don't worry about it!
Good old (round) brackets are nearly always good enough!
Chris