After the disputes we learned between Beis Shammai and Beis Hillel as to whether a person's utensils and possessions may perform work on his behalf on Shabbos, our Mishnah takes up a parallel question: work and trades performed by a non-Jew on behalf of a Jew on Shabbos - is there a problem with that?
The text of the Mishnah:
"Beis Shammai omrim: ein nosnin oros le'abdan velo keilim lekoves nachri, ela kedei sheye'asu mib'od yom. Uvechulan Beis Hillel matirin im hashemesh".
The view of Beis Shammai:
"Ein nosnin oros le'abdan" - one may not hand over hides to a professional tanner.
"Velo keilim lekoves nachri" - and one may not hand over garments to a non-Jewish launderer.
"Ela kedei sheye'asu mib'od yom" - unless there is enough time for the work to be completed during the day, before Shabbos begins.
It is obvious that a person may not hand a non-Jew work to be done for him on Shabbos itself; here, however, the non-Jew is able to do the work before Shabbos, on Shabbos, or after it. In the view of Beis Shammai, it is incumbent upon the Jew to make certain that the non-Jew will do the work before Shabbos, and one may not leave it to him to do it on Shabbos even if that is convenient for him.
The view of Beis Hillel:
"Uvechulan Beis Hillel matirin im hashemesh" - in all of these cases Beis Hillel permitted handing the work to a non-Jew even right up until sunset. And even though the non-Jew may well do the work on Shabbos itself - for example, where it was arranged with him to do it on Sunday and he chooses to do it on Shabbos - there is no prohibition in this according to Beis Hillel. According to Beis Shammai, however, this is a problem, and the one who hands over the work must make certain that it will be completed before Shabbos begins.
In summary: This Mishnah continues the discussion between Beis Shammai and Beis Hillel, this time regarding handing work to a non-Jew - hides to a tanner and garments to a launderer. Beis Shammai require that the work be done while it is still day, whereas Beis Hillel permit handing it over even as the sun is setting, and the Jew is not required to prevent the non-Jew from doing it on Shabbos.