There are different ways to write in English. You have indeed found the cursive form of the letter "s", and cursive is a whole separate way of writing letters that resemble the "printed" English letters. Cursive was commonly used in past documents but is not very prominent today. Cursive is no longer taught in many schools but some still do use it for essays and such (though typing is much preferred to cursive). Today, cursive is dying out. While it's worth a look, you don't need to memorize the cursive letters.
It appears that some states are trying to bring cursive back. And yes, older generations still use cursive (I think that a lot of people born after 2000 don't commonly use it, but I'm just basing this off of my own observations of myself and the people around me).