Welcome to Maseches Shabbos in Seder Moed. We begin with Chapter 1, Mishnah 1. Most of the tractate as a whole deals with the melachah of hotzaah - transferring objects on Shabbos: both transferring from a private domain to a public domain and from a public domain to a private domain, that is, from a private area to a public area or the reverse, and also carrying four amos within a public domain. Both are Torah-level melachos, and in the course of the discussion we will also deal with the various rabbinic prohibitions.
The first Mishnah deals with carrying out and moving objects, and it opens with the words: "Yetzi'os haShabbos" - the going out and the carrying out from a private domain to a public domain on Shabbos.
The meaning of the term "yetzi'os":
When the Mishnah uses the language of going out or carrying out, it does not mean only carrying out from a private domain to a public domain, but also bringing in - bringing an object from the public domain, the public area, into the private domain, the private area. Since bringing in is also included in "going out," the Mishnah uses this term in a general sense.
Accordingly, the Mishnah lists four categories of cases:
"Shtayim shehen arba bifnim" - two Torah-level melachos for the person standing inside, plus two more that are added to them and are prohibited only rabbinically.
"U'shtayim shehen arba bachutz" - two Torah-level melachos for the person standing outside, plus two more whose prohibition is rabbinic.
"Keitzad?"
The Mishnah presents the matter through the scenario of a homeowner giving charity - a loaf of bread or something similar - to a poor man standing outside. The homeowner is the man inside, and the poor man is the man outside, and all the cases are discussed in this context.
The cases that carry Torah-level liability:
The Mishnah begins: "He'ani omed bachutz u'ba'al habayis bifnim" - the poor man stands outside and the homeowner is inside.
"Pashat he'ani es yado lifnim v'nasan l'soch yado shel ba'al habayis" - the poor man extends his hand with the object (such as a basket he wants the homeowner to fill with food) into the private domain, and places the object into the homeowner's hand. He has thus performed both parts of the melachah: the akirah - lifting the object from the place where it had been resting, and the hanachah - setting it down into the homeowner's hand, which is inside the private domain.
"O shenatal mitochah v'hotzi" - the poor man took from the private domain, for example from the homeowner's hand, carried the object out to the public domain and set it down.
"He'ani chayav u'ba'al habayis patur" - in both directions the poor man is liable, since he performed the entire melachah himself, and these are the two melachos of the one standing outside. The homeowner is exempt, since he did nothing.
"Pashat ba'al habayis es yado lachutz v'nasan l'soch yado shel ani" - here the homeowner is the one who acts first: he takes the food item (akirah), extends his hand outside and places it into the poor man's hand (hanachah). He has therefore performed the entire melachah of carrying out by himself.
"O shenatal mitochah v'hichnis" - the homeowner took from the poor man's hand, for example the basket, and brought it into the private domain.
"Ba'al habayis chayav v'he'ani patur" - in both cases the homeowner is liable, one for carrying out and one for bringing in, and the poor man is exempt since he did nothing. These are the four cases that carry Torah-level liability: one of carrying out and one of bringing in for the homeowner, and one of carrying out and one of bringing in for the poor man.
The cases whose prohibition is rabbinic:
"Pashat he'ani es yado lifnim v'natal ba'al habayis mitochah" - the poor man extends his hand into the private domain, and the homeowner takes from his hand and sets the object down. The poor man performed only the akirah - he lifted the object outside and brought it inside; and the homeowner performed the hanachah. The two of them thus divided the melachah between them.
"O shenasan l'sochah v'hotzi" - the homeowner lifts an object and places it into the poor man's hand, and the poor man carries it out.
"Sheneihem p'turin" - both are exempt.
"Pashat ba'al habayis es yado lachutz v'natal he'ani mitochah" - here the homeowner is the one who begins and extends his hand outside: he took an object from the private domain and brought it out to the public domain, but did not set it down; rather, the poor man takes it from his hand and sets it down.
"O shenasan l'sochah v'hichnis" - the poor man places an object into the homeowner's hand, and the homeowner brings it inside. The poor man therefore performed the akirah and the homeowner the hanachah.
"Sheneihem p'turin" - both are exempt.
In summary: in the last four cases both are exempt by Torah law, but they transgress a rabbinic prohibition, since each of them performed either an akirah or a hanachah - only that he did not perform both. And by performing only an akirah or only a hanachah, a person transgresses carrying out or bringing in on a rabbinic level.