1800s or 1800's?

It depends on the context.

1800s is talking about a group of years.

1800's is possessive and is usually used to refer to ATTRIBUTES of that century.

No apostrophe is needed.

It is only used in a date when it is showing possession, such as, "An 1800's hair style."

An apostrophe takes the place of something that is missing. It is never correctly used to show a plural form.

I am ready = I'm ready. (Correct.)

He is walking to school = He's walking to school. (Correct.)

He is kicking some tennis balls = Hes kicking some tennis ball's. (WRONG! Twice over!)

So the short answer is that 1800's is wrong (in this instance) and 1800s is correct.

If, however, you were talking about the fashions of that period, you could say "In the 1960s short skirts were all the rage, but 1860's skirts were longer."

An apostrophe is used to show the possessive case, where the missing part is "his" - so that "Bill, his book" becomes "Bill's book". Note that it would not become "Bill's book's" unless you wanted to describe "Bill's book's cover" . . .

I hope I haven't managed to confuse you more than ever!

1800s. It s a plural, like teens or twenties. No need for the apostrophe.

1800's is proper.