This chapter deals with the laws of milah on Shabbos - a bris milah that falls on Shabbos. We begin with Mishnah 1.
Rabbi Eliezer's Position - Bringing the Knife:
Rabbi Eliezer says: if a person did not bring the instrument - the knife or scalpel used for the bris milah - to the place where the milah is to take place before Shabbos, he may bring it on Shabbos, and even through a public domain. However, he must carry it "meguleh" - uncovered, so that everyone can see what he is carrying in his hand. The reason: to show all that the mitzvah of milah is precious and dear to us, and that this is Shabbos.
In a Time of Danger:
When the gentiles decreed against the Jewish people that they may not circumcise, and it was impossible to carry the knife openly through a public domain, he covers it, so that the enemies will not see what he is carrying and will not realize that he is bringing a circumcision knife. But the covering is done "al pi eidim" - in the presence of witnesses: Jewish witnesses who see, before he covers it, that what he is covering is a circumcision knife. That way they will know that he is not carrying something forbidden to be carried on Shabbos, and other Jews will not suspect him of desecrating Shabbos.
The Scope of Rabbi Eliezer's Leniency:
Rabbi Eliezer holds that not only are those activities permitted on Shabbos which by their very nature must be done on Shabbos, but even activities that could have been done before Shabbos - for these too one may desecrate Shabbos. After all, the knife could have been brought while it was still day, and nevertheless, if he forgot or simply did not bring it, Rabbi Eliezer permits bringing it on Shabbos.
Rabbi Eliezer goes even further, and takes us back to the very beginning of the entire process:
Chopping wood.
Making coals from it.
Smelting the metal with them.
Fashioning a knife out of it.
Even if none of this was done before Shabbos, it may all be done on Shabbos, since in his view all the preparations needed for the milah are permitted on Shabbos.
Rabbi Akiva's Rule:
Rabbi Akiva disagrees and establishes a rule: "kol melachah she'efshar la'asosah me'erev Shabbos - einah dochah es haShabbos" - any labor that could have been done before Shabbos does not override Shabbos. Even if this will prevent the bris from taking place - he forgot to bring the knife and will not be able to circumcise on Shabbos - since this labor was possible before Shabbos, no desecration of Shabbos may be permitted for it. The Torah does not permit it.
What, then, does override Shabbos?
The Mishnah continues and teaches that only those activities are permitted which by their very definition cannot be done before Shabbos: the act of circumcision itself and the procedures that follow immediately afterward. Since the mitzvah applies only on the eighth day, and when the eighth day falls on Shabbos the Torah says that the milah is performed on Shabbos, it follows that these things, which there is no way to do before Shabbos, override Shabbos. But any activity that could have been done before Shabbos does not override it.
In summary: in this Mishnah, Rabbi Eliezer and Rabbi Akiva disagree about the scope of the labors permitted for the sake of milah on Shabbos. According to Rabbi Eliezer, even preparations that could have been done while it was still day - bringing the knife, and even chopping the wood and fashioning the instrument - are permitted on Shabbos; whereas Rabbi Akiva rules that any labor that could have been done before Shabbos does not override Shabbos, and only activities that could not have been done beforehand override it. We also learned the law of bringing the knife uncovered, and the law of covering it in the presence of witnesses in a time of danger. And of course, we rule like Rabbi Akiva.