Placing a tablecloth on the table: It is customary for there to be a tablecloth spread over one’s [dining] table throughout the entire Shabbos. One may not swerve from this custom. There are those who have the custom to spread two tablecloths over the [dining] table [and so is the widespread custom today]. [It suffices to use a white tablecloth and a plastic over it in order to fulfill this custom, however some use two white tablecloths as was done in past times prior to the plastic covering being used.]
Q&A on the tablecloths
Should one cover other tables of the house?
One is to cover all the tables that are within the room that the meal is eaten in. It is likewise proper to cover the tables of the other rooms of the house.
What color tablecloth is one to use?
The age-old custom is to use a white tablecloth to cover the dining table.
When after Shabbos may the tablecloth be removed from the table?
Some write one is not to remove the Shabbos tablecloth until after Havdala.
Covering the tables in Shul:
It is customary to cover the tables in Shul with a white tablecloth.
 Likkutei Maharich; See Q&A regarding other tables of the house.
 See previous footnotes
 Admur ibid; M”A 262 in name of Rashal and Mateh Moshe
This is besides for the cloth which covers the bread. The reason for this custom is because they want to ensure that the table remains constantly covered throughout the entire Shabbos, and hence even when they need to shake the [upper] tablecloth to remove crumbs from it, the table will still remain covered. [Admur ibid] There is however no Ohel prohibition involved in having only one tablecloth which one shakes and replaces on Shabbos, as is explained in Vol. 2 “The Laws of Ohel”. [Admur in 315 and M”A 262:1 unlike Bach]
 See Piskeiy Teshuvos 262 footnote 2
 Biur Halacha 262 “Yisader”, brought in Ketzos Hashulchan 73 footnote 13; Elya Raba 262:3; Peri Megadim 262 M”Z 1; Makor Chaim 262; Likkutei Maharich
 Aruch Hashulchan 262:2 rules that all the tables of the house are to be covered throughout the entire Shabbos, and it is considered a great shame for the Shabbos if the table becomes uncovered. However, see Kaf Hachaim 262:22 which brings that there is no need to cover the other tables of the house which one is not eating on, as the Divine blessing only rests on the table which one eats on and says Birchas Hamazon.
 Drashos Mahariy Even, student of Rashba; See Maharshag 2:82
 M”B 262:4 in name of Elya Raba and Aguda
 Makor Chaim 271:9