Parentheses are punctuation marks used to set aside information that may be useful to the reader, but not essential. Interjectory information like explanations/clarification, comments, qualifiers, and appositives are appropriate phrases to put inside parentheses. Information enclosed in parentheses gives further detail to a sentence, but when reading, can be totally skipped without negatively impacting the sentence outside of the parentheses. Essentially, you should still have a complete thought/idea/sentence whether the reader chooses to read what’s in the parenthesis or skim over that portion.
Ex. There are five (5) shipping options for your order. –clarification
Ex. The forecast called for rain (but I’ll believe it when I see it). -comment
Ex. You will face (very) harsh consequences for failing to meet the deadline. –qualifier
Ex. Mr. Hart (the author) will be giving a special lecture tomorrow. –appositive
In each of the above sentences, you can skip the portion in parentheses and still understand the sentence. You simply have fewer details about the matter.
When using parentheses, it is important to remember to place the punctuation of the complete sentence outside of the final parenthesis unless the entire sentence is within both parentheses.
The forecast called for rain (but I’ll believe it when I see it).
(The forecast called for rain but I’ll believe it when I see it.)
Also, try not to use parentheses within parentheses. That gets confusing. Find another way to say it.
As for the strange parentheses and/or/maybe apostrophes in the image, neither parentheses nor apostrophes are appropriate for what the sign says (whatever those marks are supposed to be). For more discussion on the image, see Self-Editing Tip #10–Apostrophes.