The "grocer's apostrophe" or "grocers' apostrophe" is what's incorrectly added to the plural form like what you're describing, the short answer is yes. '60s not 60's.
Technically speaking you use single quotes there, but not apostrophes. However I mention it because the quotation mark that you used is a single quote.
To form possessives of nouns
To show the omission of letters
To indicate certain plurals of lowercase letters