One of the fun parts of Rosh HaShanah is Tashlich, a great little ritual that lets one deepen the holiday experience and take a nice walk with the family all at the same time. It's a nice afternoon out for my family, especially since the city we live in reecently renovated the local park at the end of the street.
The purpose of Tashlich is simple: walk to a body of water, chuck some bread or other disposable food item into it and recite some verses. What's more, there is a tremendous amount of variability when it comes to exactly what is done as part of the ritual. The Artscroll Rosh HaShanah machzor has a section that rivals Musaf in length while the Rinat Yisrael siddur from Israel has 2 pages or so dedicated to it.
However, there is one thing that everyone agrees on: one may not do Tashlich on Shabbos. Should the first day of Rosh HaShanah fall on Shabbos, it is understood that Tashlich is to be performed on the 2nd day. But when one examines the ritual, one must wonder why.
To thicken the plot, the oldest mention of the ritual in the form we know it today comes from the Maharil around 600 years ago although there are cryptic early mentions as the link above notes. Although it is a popular custom to throw bread into the water for different potential reasons, some authorities oppose the custom. Interestingly the Maharil notes that he opposes the throwing of bread into the water on Shabbos which means that in his time folks did perform Tashlich on Shabbos. If that's the case, what happened to change that and forbid it?
The short answer is: Artscroll. Well okay, not Artscroll but as mentioned above, they do seem to have the longest Tashlich service around (at least in English-Hebrew, I haven't perused any all-Hebrew kabbalistic machzorim to determine if there are longer versions).
What seems to have happened is as follows: in the beginning the Tashlich service consisted of the recitation of Micah 7:18-20. Being as it was a relatively short passage, it was easy to memorize. A person could do so and then go down to the river/lake/sea, recite it and go home.
Over time the ritual grew, probably as a result of people showing up and saying "Wait, that's it? We need to say more!" (A similar phenomenon happened with Bircas HaChamah which started as a single blessing and turned into a multi-page service) Ultimately it became important to learn the Tashlich service by heart which meant shlepping your machzor with you to the water's edge and that creaetes problems on Shabbos if you don't have an eiruv. This presumable led to the custom to avoid Tashlich on Shabbos even in those places that did have an eiruv. Minhag Yisrael Torah hi, after all. Therefore nowadays even if you live in a place where you can carry on Shabbos you still do Tashlich on the 2nd day.