Before us is Mishnah 1 in chapter 3 of Maseches Kiddushin. The Mishnah deals with two central topics: kiddushin performed through an agent, and kiddushin that depend on time - that is, kiddushin which are said to take effect after thirty days, or from now and after thirty days.
The first case: an agent who betrothed the woman to himself:
"Ha'omer lachaveiro: tzei vekadesh li ishah plonis, vehalach vekidshah le'atzmo - mekudeshes" - one who says to his friend: go and betroth a certain woman for me, and the friend went and betrothed her to himself, instead of betrothing her on behalf of the sender. The halachah is that the kiddushin are valid, and she is betrothed to the agent.
We do not say that since she knew nothing of the agency - that in fact this man had come to represent the sender - perhaps she wanted specifically to be betrothed to the sender, and it turns out that he misled her. We do not make such a claim; rather we say that he went and betrothed her, and it makes no difference that another man originally sent him on a mission.
Nevertheless, the Gemara notes that although what he did is done and the kiddushin take effect, the agent acted improperly and is called a swindler.
The second case: "after thirty days":
"Vechen ha'omer la'ishah: harei at mekudeshes li le'achar sheloshim yom, uva acher vekidshah besoch sheloshim yom - mekudeshes lasheni" - a man betrothed a woman on condition that the kiddushin take effect only after thirty days, and within that period another man came and betrothed her. The halachah is that she is fully betrothed to the second man, while the first kiddushin never took effect at all: they had no moment in which to take hold, since before the thirty days arrived another man preceded him and betrothed her.
The practical difference concerns Terumah: if the second man, whose kiddushin took hold, is a kohen and she is the daughter of a Yisrael who does not eat Terumah - "ocheles biTerumah" - she does eat Terumah, since she is the kohen's full wife in every respect.
The third case: "from now and after thirty days":
Here we are dealing with one who told a woman that she is betrothed to him "me'achshav ule'achar sheloshim yom" - from now and after thirty days. This wording is unclear, and it can be understood in two ways:
A condition: he means that the kiddushin take effect retroactively from this moment, except that their taking effect is conditional on a wait of thirty days. Once the thirty days pass, the kiddushin take effect retroactively from the original time.
A retraction: at first he said "from now," and then he retracted and said "after thirty days" - meaning that he wants the kiddushin to take effect only at the end of the thirty days.
Therefore, when another man came and betrothed her within the thirty days, a doubt arises, and she is "mekudeshes ve'einah mekudeshes" - betrothed and not betrothed: perhaps she is betrothed to the first, since at the end of thirty days his kiddushin take effect retroactively and it makes no difference that the second betrothed her in the interim; and perhaps he truly retracted, and this is like the previous case in which she is not betrothed until after thirty days, and the second man came and preceded him.
Because of this doubt the Mishnah rules: "Bein bas Yisrael leKohen, bein bas Kohen leYisrael - einah ocheles biTerumah" - whether she is the daughter of a Yisrael betrothed to a kohen, or the daughter of a kohen betrothed to a Yisrael, she does not eat Terumah. Both sides are equal in this:
The daughter of a kohen where one of those betrothing her is a Yisrael: even though she previously ate Terumah, there is now a doubt that perhaps she is married to a Yisrael and has thereby lost her right to eat Terumah.
The daughter of a Yisrael where one of those betrothing her is a kohen: she does not eat Terumah, since it is possible that she is not married to the kohen but to the Yisrael.
In summary: in this Mishnah we learned three rulings: an agent who betrothed to himself the woman he was sent to betroth for the sender - his kiddushin are valid, even though he acted deceitfully; one who says "you are betrothed to me after thirty days" and another man came and betrothed her within that time - she is fully betrothed to the second, and if he is a kohen she eats Terumah; and one who says "from now and after thirty days," whose wording is in doubt between a condition and a retraction - she is betrothed and not betrothed, and out of doubt she does not eat Terumah, whether she is the daughter of a Yisrael betrothed to a kohen or the daughter of a kohen betrothed to a Yisrael.