Before us is Mishnah 2 in the fourth chapter of Maseches Kiddushin. In this Mishnah the Sages define two of the categories mentioned in the previous Mishnah: a shesuki and an asufi.
Who is a shesuki:
"Kol shehu makir es imo v'eino makir es aviv" - a child who knows who his mother is, but does not know who his father is. The name "shesuki" comes from the fact that his mother silences him: when he asks to know who his father is, she tells him to be quiet, since she will not reveal who his father is. Because the identity of the father is unknown, we are concerned that he may have been a mamzer, and therefore the child is a possible mamzer.
Who is an asufi:
"Kol shene'esaf min hashuk, v'ein makir lo lo es aviv v'lo es imo" - an infant who was found abandoned and was gathered in from the street, and neither his father nor his mother is known. Since he was abandoned, there is reason to be concerned that he came from a forbidden relationship.
Torah law and a rabbinic safeguard:
The Gemara explains that by Torah law a shesuki and an asufi are permitted to enter the congregation and to marry an ordinary Jew. The basis for this is the exposition of the verse "Lo yavo mamzer bikhal Hashem" - it is specifically a definite mamzer who is forbidden to enter the congregation, but a possible mamzer is permitted. It was the Sages who established a higher standard in matters of lineage - they elevated the status of the congregation of Israel and instituted a rabbinic law that one whose status is in doubt may not marry an ordinary Jew.
From that same verse we also learn the other side: a definite mamzer is permitted to marry a possible mamzer. "Lo yavo mamzer" - into a congregation that is definite; a mamzer may not marry someone who is certainly part of the congregation of Israel, but someone whose status is in doubt is permitted to marry him.
The opinion of Abba Shaul:
The Mishnah concludes with the opinion of Abba Shaul, who would call a shesuki a "beduki" - that is, one who can be examined. He means that in his view one may ask the mother who the father is, and she is believed. If she said that a Jew of proper lineage had relations with her and he is the child's father, the child is of proper lineage and is no longer considered a possible mamzer.
In summary: This Mishnah defined a shesuki - one who knows his mother but does not know his father, and an asufi - one gathered in from the street, whose father and mother are both unknown. Both are possible mamzerim, and by Torah law they are permitted to enter the congregation, since "lo yavo mamzer" was said only regarding a definite mamzer; it was the Sages who established a higher standard in matters of lineage and forbade them. We also learned that a definite mamzer is permitted to marry a possible mamzer, and the approach of Abba Shaul, who called a shesuki a "beduki," since the mother is believed to say who the father is and thereby establish the child's proper lineage.