There's a nice map of it in Swedish wikipedia where you can see how big it is: https://sv.wikipedia.org/wiki/Stockholms_sk%C3%A4rg%C3%A5rd
There's a similar archipelago outside of Gothenburg too which is very easily accessible. These areas have great importance as recreation areas and generally in Swedish culture, which is why we thought this was an important word to include in the course.
It's incorrect to use the apostrophe for the genitive in Swedish, although you can certainly find lots of cases where people do anyway (hint: look at the names of hair salons and the like).
Exception: You may use an apostrophe for names that end in an s sound if it's absolutely necessary for clarity (Mats - Mats') but you shouldn't normally use the apostrophe there either.
Rereading my old answer here made me a bit confused. I'm not sure what the original question was really about, it looks like I might have answered a a different question. Anyway if you get this as a translation exercise, of course you should type Stockholm's archipelago with an apostrophe in English, but if you get it as a listen and type in Swedish, you must write Stockholms skärgård since it's incorrect to add an apostrophe in Swedish.
In English, you could also write Stockholm archipelago, with or without 'the', we accept both + versions with 'of'. I don't think anyone ever calls it "Stockholms archipelago" without an apostrophe in English.