The only fast day that is observed on Shabbos when they coincide is Yom Kippur. When Yom Kippur falls on Shabbos (as it does this year) we do not eat or drink the entire Shabbos. In doing so we are in effect being mevatel the obligation of Kiddush on Shabbos, along with the other mitzvos involved in eating on Shabbos. Nevertheless, we “trade” these, and refrain from eating and drinking as is prohibited on Yom Kippur.
There has been a question raised by the Acharonim, which attempts to facilitate the mitzvos of both Shabbos and Yom Kippur. There is a halachah of tosefes Shabbos. This means that one can add onto Shabbos, both before and after the actual time when Shabbos is to be brought in and leave. According to some opinions one can make Kiddush and eat his seudah during this added time. Based on this, the Acharonim ask, why don’t we bring in Shabbos early, and make Kiddush and eat a seudah before the zeman that Yom Kippur will start? That way we will fulfill the mitzvos of Shabbos and still keep the prohibitions of Yom Kippur.
Some Achronim say that since we can fulfill the obligation of Kiddush by reciting the Shemoneh Esrei, this is sufficient. Others say, based on a Tosafos Rid in Pesachim 106a, that when Shabbos falls on Yom Kippur there is no obligation to recite Kiddush at all. The only time that there is an obligation to recite Kiddush is when there will be a seudah. Since when Shabbos falls out on Yom Kippur there is no seudas Shabbos, there is also no obligation to recite Kiddush.
This has ramifications regarding one who is required to eat on Yom Kippur for health related reasons. Must he recite Kiddush before eating or not. Rav Akiva Eiger in Shulchan Aruch, Orach Chaim 618, rules that he must recite Kiddush. Perhaps if an individual must eat then he has an obligation to recite Kiddush. For all those who will not be eating on Yom Kippur, they do not have an obligation to recite Kiddush in the first place.
Others suggest that one cannot separate Shabbos from Yom Kippur by accepting Shabbos early. This is because when one accepts Shabbos early he is declaring it night and it is thereby considered the next day. That means that it will also be Yom Kippur. This is a matter of dispute that has many applications. One example is whether one can make a bris milah during that time. If for some reason the baby was unable to have a bris until then and Friday was the eighth day, once one accepts Shabbos is it considered the next day and they can no longer perform the bris or is it still the eighth day and they can perform the bris. Another example of this is when Rosh Hashanah falls out on Friday and there was no shofar available until after they accepted Shabbos. May they still blow shofar or is it considered the next day? For more on this topic see Taz, Orach Chaim 668.
The Rambam says (Hilchos Avodas Yom Hakippurim 4:1) that when Yom Kippur falls out on Shabbos the kohen gadol must bring the korban Mussaf of Shabbos. We know that the avodah of Yom Kippur must be done exclusively by the kohen gadol. However, why does the Rambam say that the kohen gadol must perform even the korban Mussaf of Shabbos? That is not part of the Yom Kippur avodah. Any kohen should be able to perform it.
The Ohr Samayach, Rav Meir Simcha of Davinsk, says that it is implied in the Rambam that when Yom Kippur falls on Shabbos, even the kedushah of Shabbos is infused together with Yom Kippur. Therefore even the korban Mussaf of Shabbos becomes part of the Yom Kippur avodah and is required to be performed by the kohen gadol.