Here we'll talk about a Kohen Gadol (High Priest) or a king who sinned after they left their position. The Mishnah says: You have a Kohen Gadol who left his position and then committed a sin for which a Kohen Gadol would have had to bring a Par Kohen Mashiach (a bull), or another case: a king who, after leaving his position, committed a sin for which a king would normally have to bring a Chatat (sin offering) of a Nasi (king).
Halakha (Jewish law) is as follows. Kohen Mashiach mevi par, the anointed Kohen, the Kohen Gadol. He has to bring a bull. He has to bring his Par Kohen Mashiach. Even though he was removed from his position, he is still considered to have the sanctity of a Kohen Gadol, and the laws of Par Kohen Mashiach still apply. But the king who lost his status as a king is looked at as an ordinary person. Once he's removed from his throne, he doesn't have that special status. Therefore, if he sins, the offering brought by an ordinary person, which is a regular Chatat, would be what he brings.