As pointed out , the source is in Chagiga 27a.
Rashi and Tosafos both say that the reason being that one does Hachnosas Orchim at one's table.
שולחנו מכפר עליו - בהכנסת אורחין:
One's table atones - by inviting guests
שלחנו של אדם מכפר. דגדול כח הלגימא כדאיתא בהגדת חלק (סנהדרין דף קג.):
One's table atones. Because the power of Legima is great, as we see in Sanhedrin 103b
Note that Tosafos in the printed version points to Sanhedrin 106a - a typo
If we go to Sanhedrin 103b we learn:
א"ר יוחנן משום רבי יוסי בן קסמא גדולה לגימה שהרחיקה שתי משפחות מישראל שנאמר (דברים כג, ה) על דבר אשר לא קדמו אתכם בלחם ובמים
Rabbi Yoḥanan says in the name of Rabbi Yosei ben Kisma: Great is Legima, as it distanced two clans from the Jewish people, as it is stated: “An Ammonite or a Moabite shall not enter into the assembly of the Lord…because they met you not with bread and with water” (Deuteronomy 23:4–5).
So we see that Legima is offering people food.
Putting it all together, I would propose that any location where one actually hosts guests has the atonement value of a Mizbeach. (This seems clear from Rashi; from reading Tosafos one could argue that even if it only has the potential to host guests but one hasn't yet done so, it would still qualify.)
I would also propose that any type of food being served would qualify, as the Ammonites and a Moabites were not expected to serve full meals; they were expected to be hospitable.
So to answer your #1: When? Once you've hosted guests at it, it has this special status.
And #2: Any surface - once it was used for hosting guests has this special status.
But I cannot find any concrete proof - just reading between the lines.