G‑d commands us (Exodus 20:8), "Remember the Sabbath day to sanctify it." This verse teaches us that one should sanctify the seventh day of the week with a verbal statement, acknowledging the Sabbath with words of praise that reflect its holiness. This ritual is referred to as kiddush (pronounced kee-doosh), which translates as sanctification.
Technically, one can fulfill the biblical commandment of sanctification by simply reciting the words of kiddush. However, our sages instituted the recitation of the prayer over a cup of wine, since the Bible often associates “remembrance” with wine.
On the holiest day of the year, The Day of Atonement, Yom Kippur, adults and children over the age of Bar and Bat Mitzvah (boys over the age of 13 and girls over the age of 12) are biblically commanded to abstain from eating and drinking. Therefore, even when Yom Kippur falls on the Sabbath, we do not recite the kiddush over wine. Nevertheless, we are still able to fulfill our obligation by remembering the Sabbath during the Yom Kippur services. Thus, while praying, one should have in mind the intention to also fulfill the obligation of sanctification.
If a person is severely ill and cannot fast, he or she still should not recite kiddush, since the sages did not institute kiddush over wine on this day. But, if an ill person ate or drank an amount (over .61 ounces) that requires the Grace after Meals (or its shorter version, see Blessings After Other Foods), they should include the standard additions for the Sabbath, as well as the yaaleh veyavo prayer, which is added to the Grace on holidays.