One of the five things that a person may not do on the day of Yom Kippur is called Tashmish Hamitah (marital relations). A person may not conduct marital relations; it is one of the five physical things that a person has to withdraw from. The Torah says that inui: that a person has to cause affliction to themselves. We see that the word inui is associated with withholding physical intimacy. In the case of Yaakov and his wives, Lavan tells him, “If you will afflict my daughters,” meaning that if he withholds intimacy from them, that will be considered an affliction. So withholding intimacy is a way that we afflict ourselves, and that's one of the interpretations of afflicting ourselves on Yom Kippur: to withhold marital relations.
But it's not only intercourse that's not allowed. The Hachamim (sages) go as far as to say that a person may not sleep in the same bed with his wife: the beds have to be separated, and they should not have any kind of intimate physical contact with each other during the course of the day of Yom Kippur. There's some debate about whether or not we treat a person's spouse as though they are completely niddah (ritually impure woman) during Yom Kippur, meaning they can't even pass objects hand to hand, can't even sit on each other's bed, and so on. So Hacham Ovadia writes that ideally a person should conduct themselves that way and should keep in mind that they should treat their wife as though she were niddah and maintain complete separation during the course of Yom Kippur. But he says that if there's a need for something, one could be lenient on some of the harchakot, the normal distancing that we make between husband and wife during the time of niddah, if necessary, as long as there's no intimacy between the two.