The letter you are referring to is called a Long S, which was derived from the old Roman cursive medial s (an s in the middle of a word).
In many alphabets, some letters take different forms depending on their position within the word. English for example has capital letters which only appear in certain situations (beginning of sentences, proper nouns, abbreviations, etc). Hebrew and Arabic have final forms of certain letters, which only are used at the end of a word. The Greek letter sigma has three forms, capital Σ, medial σ, and final ς.
The long s was used for a very long time, and can be seen in many ancient manuscripts. It eventually fell out of use in the 19th century, first being abandoned by printers and typewriters, and then falling out of handwriting shortly afterwards.
The long s originated as a medial s (in the middle of a word). So you could consider it to be a "middle form" of the letter s. It occurred at the beginning and in the middle of words, but rarely/never at the end of words. So, for example, "sasquatches" would be written "fafquatches", because the "normal s" was actually the final form of the letter. In words with double s, like Mississippi, the long s would precede the short s, thus "Mifsifsippi".
Basically, it was the exact same letter, just a typographical form specifically for use within the word, not at the end. The same way we use capital letters at the beginning of sentences.
What you're talking about is typographically called the "long S." The long S looked like an F, but usually without the cross-stroke or with only a very short cross-stroke; it was used initially (at the start of a word) and medially (inside a word), while the short "s" was used terminally (at the end of the world). In other words, "sentiments" would look something like "fentiments"; I can't reproduce the long S on the computer.
The origins of this seems to be in the difference between how the majuscule (uppercase) S and minuscule (lowercase) s were handwritten.