daf bit: Zevachim 55
Different offerings have different rules for how they are offered and eaten. The olah, or burnt offering, is wholly consumed and so nobody eats any of it. At the end of yesterday's daf the mishna says the communal peace-offering and the guilt-offerings must be slaughtered in a particular place within the temple (the north) and are eaten within the temple, by male priests, a day and a night until midnight. On today's daf the mishna says that the thanksgiving offering and the ram that the nazarite brings are of lesser sanctity; they are slaughtered anywhere within the temple court, and they are eaten in any part of the city, by any person, the same day and night until midnight (except for the parts that are given to the priests). The peace-offering is also of lesser sanctity; it is slaughtered anywhere within the temple court, and it is eaten anywhere in the city by anybody for two days and one night. All of these may be prepared in any manner, in contrast to the Pesach offering (coming up in a couple pages) which must be roasted. (54b-55a)
The thanksgiving and peace offerings are unscheduled and more "donor-driven" than the set communal offerings (times specified in torah) and guilt-offerings (owed when you transgress). I have the impression that the thanksgiving and peace offerings were things you planned a family feast around, so it makes sense that you can eat them anywhere in Jerusalem with anybody you like. I don't yet know why you get an extra day for the peace-offering.

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