Mishnah 1 in Chapter 4 of Tractate Kerisos introduces the central topic for the rest of the tractate: Asham Taluy (a suspensive guilt offering). This offering is called "Taluy" (suspensive) because it suspends the consequences of a situation where a person performed an act that, had he definitely performed it, would obligate him to bring a chatas (sin offering) - but he does not know for certain whether he actually performed it. Since one does not bring a chatas out of doubt, as we have learned, he brings an Asham Taluy instead. An Asham Taluy suspends the consequences of the sin, but it does not provide complete atonement: if it is later clarified that he indeed committed the transgression, he will have to bring a chatas.
It must be remembered that a chatas is only brought for a transgression that carries liability for kares, and when it was committed unwittingly. Therefore, all the cases in our Mishnah deal with a person who thought he was acting correctly and in a permitted manner, and it ultimately became apparent that he may have stumbled in a prohibition and he may not have - and therefore he brings an Asham Taluy.
The wording of the Mishnah:
"Safek achal chelev safek lo achal" - a person who does not know if he ate chelev, which is the forbidden fat for whose consumption one is liable for kares, as we have learned. "Va'afilu achal, safek yesh bo kashiur safek she'ein bo" - and even if he ate it, he does not know if he ate the obligating measure. For all dietary prohibitions, the obligating measure is a kezayis, eaten within the timespan of k'dei achilas pras, and the main point here is the measure of a kezayis.
Two approaches to understanding the Mishnah:
Rashi, the Rambam, and the Bartenura: Here are two separate cases. The first - he ate and does not know what he ate. The second - he does not know if he ate the measure of what was forbidden to him, as there is no liability for a chatas for less than the measure.
Tosafos: Here we have only one case, which is a safek sefeika (a double doubt): first, he does not know if he ate the forbidden item; and even if he ate it, he does not know if he ate the obligating amount of it. According to this approach, a major novelty is introduced, that even for a safek sefeika one brings an Asham Taluy - but the halachah is not like this.
"Ikba issura":
The Mishnah details the case, since it is not enough for a person to be in doubt whether he committed a transgression or not. What is required is what is called "ikba issura" - that the prohibition is established and absolute in reality, meaning that we know for certain that there was a prohibited item here, and the doubt is only whether the person stumbled over it and came into contact with it.
Therefore, the Mishnah is not dealing with someone who ate one item and is in doubt if it was chelev or not, but rather: "Chelev v'shuman lefanav, achal es echad mehen v'ein yadua eizeh mehen achal" - two pieces of fat were placed before him, one was chelev and one was fat that is permitted to eat. He ate one of them and his friend ate the other, and he can no longer clarify which of them he ate. In such a case, he brings an Asham Taluy, until it is clarified who ate the chelev, if it can be clarified at all.
And similarly in the case of the measure: some learn that from the moment it is known for certain that there was chelev there, even if it is not known how much chelev was there, this is considered "ikba issura" and is sufficient to raise the question. Others learn that one must know for certain that there was a complete measure there, at least a kezayis of chelev, and the doubt is only whether the person ate the measure of a kezayis. Regardless, in all cases we are dealing with a person who ate something that he thought was permitted.
Parallel cases in other areas:
"Ishto va'achoso imo... v'shagag b'achas mehen v'ein yadua b'eizo mehen shagag" - a person whose wife and sister were with him in the house, and he cohabited with one of them and does not know with whom he cohabited. If it had been his sister and done unwittingly, he would be obligated to bring a chatas; and since his wife is permitted to him and it is not known which one of them it was, he brings an Asham Taluy.
"Shabbos v'yom chol... v'asah melachah b'achas mehen v'ein yadua b'eizo mehen asah" - this refers to the time of bein hashemashos (twilight), which has an aspect of Shabbos and an aspect of a weekday. He performed labor during bein hashemashos and does not know if the labor was performed on Shabbos or on a weekday, and therefore he brings an Asham Taluy.
The term "shagag" used here requires explanation, and one should not be too precise with it: it is used to parallel the language of the following Mishnah. After all, if he cohabited with his sister - he indeed sinned inadvertently and did this by mistake; but if he cohabited with his wife - there is no inadvertent sin here at all. The main point is that he does not know which of them it was, and therefore he must bring an asham talui.
In the case of Shabbos and a weekday, the Rishonim disagree: according to the Rambam, the law applies both during bein hashemashos (twilight) of Friday evening and bein hashemashos of Saturday night - in both of them there is a weekday and Shabbos, and it is unknown on which side of the dividing line the melachah was performed. According to Tosafos, it refers specifically to Saturday night, since there it is known with certainty that there was a prohibition - Shabbos was certainly present, and the doubt is only whether the melachah was performed during Shabbos or after it. Thus, the halachah in this case is a subject of dispute between the Rambam and Tosafos.
In summary: In this Mishnah we learned the fundamental concept of an asham talui - an offering brought for a doubtful transgression that would require a chatas if it were certain, which suspends the consequences of the sin and does not achieve complete atonement. We explored the two approaches to understanding the language of the Mishnah, the condition of "ikva isura" (the prohibition is established) which requires certainty that the prohibition exists, and the three cases listed by the Mishnah: cheilev and shuman, his wife and his sister, and Shabbos and a weekday during bein hashemashos.