EDIT Earliest use Braces { } are found in the 1593 edition of Francois Vieta's Zetetica.

Source: A History of Mathematical Notations, Cajori, vol. 1, page 391
EDIT Synonyms
North American English: braces
French: Akkolade (hug)
EDIT Use Brackets generally serve to enclose textual material and to differentiate between various additions to a sentence (inserted sentences, letters, parts of words).
Curly brackets are hardly ever used in continuous text, they are used additionally when parenthesis and square brackets are already in use. The sequence is: square brackets are used for insertions into parenthesis, curly brackets are inserted into square brackets ([{}]).
EDIT Mathematics In mathematical formulas brackets have a very strict order. The curly brackets are the outermost ones after the parethesis and the square brackets.

Also, in mathematics a set can be defined by listing its members between curly Brackets :

A = {3, 1, 4, 2,5}
B = {cyan, magenta, yellow, black}

(the other option is do describe it in words, like: »A is the set whose members are the first five positive whole numbers« or »B is the set whose members are the four colours used for printing).
EDIT Programming In computer programming, curly brackets sometimes denote the beginning and ending of a sequence of statements.
EDIT Music The curly brackets connect two or more lines of music that are played simultaneously.