Is 1960's a contraction or abbreviation? What does it mean exactly?

Answer 1

Neither. It's a plural.
Refers to the years between 1960 and 1969
Frequently ABREVIATED by saying "the 60's."
Such as: "That noted rockband of the 60's."
Note that an apostrophe can be used in some plurals of numbers and single syllable words although it seems to be disappearing from the language.

Answer 2

It's more an abberation, a bastardized shortcut. Similar errors include the Smith's The Jones's, etc The proper and correct form is; The Smiths, and The Joneses. The Nineteen-sixties respectively proper form, though many educated people tend to refer to them as 'the Sixties' . If avoiding all possiblity of error is necessary, the sixth decade of the twentieth century can leave little doubt.

I myself refer to those years as the 60's!

Answer 3

Interesting! My best guess is Collective Noun.