There are two correct ways and two incorrect ways to express decades in writing.
If you are going to write out the full decade using numbers, you should place an “s” after the numerals. For instance, you would write “the 1980s,” or “the 1850s.” You should not place an “apostrophe -s” after the decade. Apostrophes are for forming contractions or indicating possession; they are not for forming plurals of numbers. Therefore it is incorrect to write “the 1980’s” or “the 1850’s.”
You may also choose to abbreviate the decade with a two-digit number. An abbreviation means that some of the written material is removed; with decades, it’s the first two digits (the century). Because you are removing them from the date, you must place an apostrophe before the other two digits (the decade). For instance, you would write “the ’80s or “the 50s.”
You can use an abbreviation for a decade only if the context clues in the rest of the sentence ensure that there is no confusion regarding the century. For instance, “I love listening to ’70s music; disco is my absolute favorite” would work because disco was popular in the 1970s. On the other hand, if you were to write, “These are beautiful vintage clothes from the ’50s,” you would have no idea if the writer is talking about the 1950s, the 1850s, or even the 1750s, and so forth. In this example, it would be best to write out the date out entirely: “These are beautiful vintage clothes from the 1850s.”
Furthermore, remember that apostrophes are for forming contractions or indicating possession; they are not for forming plurals of numbers. Therefore you should not write “the 80’s” or “the 50’s”.
Let’s recap:
DO write out a decade with numerals and an “s” after them.
DO write a decade as a two-digit number with an apostrophe before it and an “s” after it, as the apostrophe takes the place of the century digits.
DON’T use an “apostrophe -s” at all when writing decades.
DO write “the 1950s.”
Do write “the ’50s.”
Don’t write “the 1950’s.”
Don’t write “the 50’s.”
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