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I'm sure you've all noticed documents in English from the 1700's often have 'F' where, if written now, there would be an 'S'. You can see what I'm talking about a few times in this example, like at the beginning where it says "Prayers faid" or in the date "Tuefday November 26. 1700."

What's going on with this? When did it start? When did it stop?

There's a typographical distinction between an actual f and the ſ you're referring to in the text. See for instance the difference between 'magiſtrats' and 'behalf' in the second paragraph.

The 'ſ' is a long 's'; the wiki article has a very long section on its history and decline of use.

In general, the long s fell out of use in Roman and italic typefaces in professional printing well before the middle of the 19th century. It rarely appears in good quality London printing after 1800, though it lingers provincially until 1824, and is found in handwriting into the second half of the nineteenth century" being sometimes seen later on in archaic or traditionalist printing such as printed collections of sermons.

See this Old English Alphabet for a more complete list of changes to the alphabet. And a ſ cousin, the German ß.

Anecdotally, Hungarian also has a similar but possibly unrelated sound: ssz. I say unrelated because Hungarian is a Finno-Ugric language rather than a Latin, Germanic, or Slavic one. I've no idea if the sound was borrowed, or if it evolved earlier in the Indo-European language family. I've asked in the Linguistics SE about the actual sound's history, hoping for much better answers than this one.