EDIT Earliest use Parentheses ( ) are "found in rare instances as early as the sixteenth century" says Cajori (1859-1930, a swiss-born U.S. educator and mathematician whose works on the history of mathematics were among the most eminent of his time).

Apparently the earliest work Cajori names in which round parentheses are found is General trattato di numeri e misure by Nicolo Tartaglia (c. 1506-1557) in 1556. Round parentheses occur once in Ars magna by Cardan, as printed in Opera (1663) Cajori does not indicate whether the parentheses occur in the original 1545 edition.

Cajori says that Michael Stifel (1487 or 1486 - 1567) does not use parentheses as signs of aggregation in his printed works, but that they are found in one of his handwritten marginal notes. Cajori expresses the opinion that these parentheses are actually punctuation marks rather than mathematical symbols.

Kline says parentheses appear in 1544. He presumably refers to Arithmetica integra by Michael Stifel.

Sources: A History of Mathematical Notations, Cajori vol. 1, page 390, 391, 392

 84.176.184.170  :  jsw 

EDIT Parentheses are punctuation marks and actually round brackets.
The term "parentheses" comes from the Greek word παρενθεσις and means an insertion of a word or phrase in a sentence or passage with witch it is not otherwise grammatically connetcted and from which it can be removed without disturbing the meaning of the sentence.
"The teacher, a tall man, left the classroom."

The round brackets have the purpose of marking such a parenthesis as insertion (or addition) that in some way needs to be excluded from the rest of the sentence and can also supply additional informations.
They can also be nested (which means another set (and maybe another) of brackets is put within the first set) but that's not too common and the inclosed brackets usually better be square ones.

 jsw 

EDIT History
Parentheses have been used where today a slash is more common, to indicate alternatives: "a mother)(father to take care of the child"

 jsw 

EDIT Typography
Text is put in parentheses without spaces before or after it. The bracket gets a single word space before and after it. If any punctuation marks belong to the inserted sentence, even exclamation marks or full stops, they belong into the parentheses as well. Any punctuation belonging to the original sentence remains outside and is put after the parenthesis if the insertion is at the end of the sentence.
Sometimes there is a need to manually adjust the distance between the bracket and the first letter within it because the ascenders and descenders of j, g or f might collide with it.

 jsw 

EDIT Mathematics
Parentheses indicate a change of precedences of operators.

Usually, the multiplication is done before additions: 2+3x4 = 14
But when parentheses are used, they cause anything within them to be done first: (2+3)x4 = 20

Brackets can also be nested within each other, the innermost is done first. From outside to inside, the order of brackets is parentheses, square brackets, curley brackets/braces, angle brackets. If that is not enough, the sequence starts over again with parentheses, but in a slightly larger font size.

Parentheses, with or without the combination of square brackets, can specify an interval (a set of natural numbers defined by a beginning and an end number). A square bracket indicates inclusion of and a round bracket exclusion of a number. For example: [4, 7) means all real numbers between and including 4 up to, but not including 7, which means 6.9999999...

 jsw 

EDIT Informatics
In many computer languages, parentheses contain the arguments to functions: substring($val,10,1) and can also show the start and end of lists.

 jsw 

EDIT Emoticons
Parenthesis are popular in digital media to create little pictures that convey the feelings of the writer. Some examples are:

:) like it / feeling good / smiley
:( don't like it / feeling bad
;) wink
(:-. sad / lovesick

 jsw