The bracket is often referred to as the square bracket, and your memory is correct, it looks like the trimmed off end of a square or rectangle.
The brace is often referred to as the curly brace (sometimes the curly bracket). The closest I can come to an even semi-accurate description is that it has a rounded shape at the upper and lower end, much like a parenthesis, but there is a small point pulled out at the halfway point of its height. There is a bit of recurve coming out of that point, so the whole thing is not fully "on the round" like a parenthesis is.
Brian - Windows 10 Pro, 64-Bit, Version 1903, Build 18362
Many of the insights of the saint stem from his experience as a sinner.
~ Eric Hoffer