A parenthetical expression is a phrase or clause that’s inserted within—in effect, it interrupts—another phrase or clause. The larger structure is complete without the smaller structure, which could be an adverb clause, as in the following four examples, or an added comment or remark that has no syntactic function in the clause. I’ve italicized the parenthetical expressions, including the one in the first sentence in this paragraph. Note that the expressions are enclosed in (surrounded by) pairs of punctuation marks: commas or parentheses (round brackets), or dashes:
Strawberry jam, for instance, doesn’t make a good spaghetti sauce.
Uncle Charlie, when he was told about the escaped fleas, broke out in a blush.
Cobras, although they are essentially moody, like an occasional chuckle.
The tone of her letter, however, brought tears to Fang’s eyes.
It’s important to note that the commas that surround phrases and clauses that are inserted between backbone functions don’t violate the rule that no punctuation can separate subjects and finite verbs: the punctuation “belongs” to the parenthetical expression:
Fang, in fact, was furious.
But, of course, Miss Hiss was amused.
Jumbo, it seems, prefers peanuts in soy sauce.
Parentheses (or round brackets) set off material that is inserted as an explanation or comment. The idea is to add information rather quietly (a brief definition or comment, for example), so as not to distract the reader from the rest of the sentence:
Uncle Charlie (a reformed gargler) met Aunt Min (at that time a mouthwash therapist) at a poltergeists’ convention.
Charlie was looking for a dealership in rubber adverbs (used in very tight clauses), but he would have willingly fallen back on wooden appositives (a boon for the unimaginative essayist).
When I can’t think of any appropriate silly examples (for instance, after a strenuous evening with the Boa Brothers), I try standing on my head.
In the sentence about Uncle Charlie and Aunt Min, no punctuation follows either parenthesis because none would have been used without the parentheses. In the second and third sentences, the commas that would ordinarily follow adverbs and examples come after the closing parenthesis marks. The rule is that the expression in parentheses should come immediately after the structure it comments on, before any other punctuation.
A pair of dashes is used to emphasize parenthetical expressions more forcefully:
Monty the Python—a snake in the grass if I ever saw one—often pretends to have fang aches.
Orange-nosed attack frogs—there’s one on your shoulder now—defeat their enemies by sucking out their wits.
If I’m allowed to write a serious example sentence for once—I’ll probably be lynched for it—I’d say that it’s best to use dashes sparingly.
Jack was injured by a booby-trapped plum—or so he claims.