A parenthesis is a word or a phrase that is inserted into a sentence to add information or explain something.
We generally put extra information inside commas.
My sister, who is called Emily, loves horses.
However, if we want the extra information to stand out more, then dashes or brackets can be used.
My sister - who is called Emily - loves horses.
My sister (who is called Emily) loves horses.
Sometimes people use the word parenthesis to refer to the brackets themselves, but strictly speaking the parenthesis is the information, not the punctuation.
The sentence should always make sense without the parenthesis, whether commas, dashes or brackets are used. Dashes are generally more informal, and are often used in emails or other casual writing.
The plural of parenthesis is parentheses.