It has become customary in kosher kitchens to designate separate tablecloths for use with meat and dairy foods. The tablecloths should be significantly different in color, size or other easily identifiable features, and easily identifiable by all members of the household as the ones used for meat or milk. (See Shulchan Aruch Yoreh Deah Siman 89:4 and Darchei Teshuva 89:48)
The tablecloths should not be used interchangeably for meat and milk without being washed, even if they have been wiped down well between uses. If, however, they were machine washed, they may be switched from meat to dairy and vice versa. (See Darchei Teshuva Yoreh Deah Siman 121:60 and Rama Orach Chaim Siman 442:11 and 451:18 and Mishna Berura there)
Some Poskim allow using the same tablecloth for meat and milk if it is flipped over and separate sides are used for each food class, since most of our foods today are served in dishes and don’t come in direct contact with the tablecloth.
However, it is best to be stringent, where possible. (See Pischei Teshuva Yoreh Deah Siman 89:18 quoting the RadVaz)