Piggul (פיגול) is an offering invalidated by improper intent. During the act of shechita (or another core service), if the kohen thinks "I will eat from this offering past its proper eating window" — meaning tomorrow, when only today and tonight are halachically allowed — the offering becomes piggul. The intent alone disqualifies. The offering need never actually be eaten late.
This mishnah examines piggul for two paired offerings — one animal, one bread — that are brought together:
Todah — the thanksgiving peace offering, brought with 40 accompanying loaves (chametz and matzah). The animal and its loaves are eaten on the day of slaughter and the night following.
Kisvei Atzeres — the two communal peace-offering sheep brought on Shavuos, alongside the Shtei HaLechem, the two loaves of chametz bread baked from new wheat. Same eating window as the Todah pair.
Both pairs raise the same two questions: when improper intent is held during the animal's shechita, does it carry over to the accompanying bread? When the intent regards the bread, does it carry over to the animal?
You have learned every Mishnah of From the Omer to Shavuos — all 53 steps.