Tables of the past were made of thick, solid wood. People would put burning hot pastries and boiling pots on them, so ḥametz dishes would occasionally spill on them. Therefore, the custom became to kasher tables by pouring boiling water over them (SA 451:20), and some were more stringent, pouring boiling water onto a white-hot stone on the table (Mahari Weil). However, today’s tables are more delicate and fragile, and would be damaged, warped, or defaced by boiling water. Since the tables are so delicate, people do not place boiling pots or burning hot pastries directly on them.
Therefore, one must clean the table well, as this kashers the table in according to its majority usage. But since sometimes hot ḥametz liquids can splash onto the table, and sometimes people place hot pastries on it, it is proper to take care not to eat at the table without a tablecloth serving as a barrier between the table and the food.
Some are scrupulous to affix nylon or contact paper, out of concern that the tablecloth placed on the table will move around, but affixing them creates a permanent barrier, over which one places the tablecloth. If it is a table on which one occasionally kneads dough, a permanent barrier must be placed or affixed to it.
For a table on which no one places hot ḥametz foods and no one kneads dough, it is sufficient to clean it well, and there is no need to cover it.
One may kasher a tablecloth on which ḥametz was eaten by laundering it in a washing machine. If a tablecloth cannot be washed, one should clean it and keep it in a locked cabinet with the ḥametz dishes.