Before us is Mishnah 2 of the seventh chapter of Maseches Yevamos, which deals with slaves that a woman brings to her husband upon marriage, and the question of whether they may eat Terumah.
A Yisrael's daughter who marries a kohen:
The words of the Mishnah: "Bas Yisrael shenisses l'kohen v'hichnisah lo avadim, bein avdei melog bein avdei tzon barzel - harei elu yochlu bis'rumah" - a woman who is not from a family of kohanim who married a kohen and brought slaves with her into the marriage: in both categories of slaves the law is the same, and all of them eat Terumah.
Avdei melog - property that belongs to the woman herself, while the husband receives only their produce. They eat Terumah because she eats Terumah, since she is married to a kohen.
Avdei tzon barzel - property that is owned by the husband. They eat Terumah because their owner is a kohen.
A kohen's daughter who marries a Yisrael:
The Mishnah continues: "U'vas kohen shenisses l'Yisrael v'hichnisah lo bein avdei melog bein avdei tzon barzel - harei elu lo yochlu bis'rumah" - a kohen's daughter who married a Yisrael no longer eats Terumah, since she married a non-kohen, and therefore her slaves do not eat either, neither of this category nor of that one.
Avdei tzon barzel - they do not eat Terumah, since they are owned by someone who is not a kohen.
Avdei melog - they do not eat Terumah, since their right to eat depends on her, and she herself does not eat because she married a non-kohen.
In summary: a slave's right to eat Terumah is determined by the one to whom the slave belongs: avdei tzon barzel depend on the husband, and avdei melog depend on the wife. Therefore, in the case of a Yisrael's daughter who marries a kohen, both categories eat, since both she and her husband eat Terumah; and in the case of a kohen's daughter who marries a Yisrael, neither category eats, since neither of them eats Terumah.