Peah, Mishnah 2. To understand this Mishnah we need to introduce an idea that is actually the subject of the next tractate - Maseches Demai. The Sages established the category of 'demai,' and it is based on the fact that the amei ha'aretz, the unlearned masses, were not careful about separating tithes - those ten percent that must be separated so that the produce is no longer tevel.
What the amei ha'aretz were suspected of, and what they were not suspected of:
Terumah gedolah - this is the first separation, and it is given to a kohen. The Gemara says that the amei ha'aretz were not suspected of failing to separate it, and therefore we may assume that terumah gedolah was separated and there is no problem with it.
First tithe and second tithe (or the tithe for the poor) - the next percentages that must be separated. Of these the amei ha'aretz were indeed suspected of not separating them.
However, leket, shich'chah and peah, as well as the tithe for the poor, are not subject to terumos and tithes. Therefore, if what an am ha'aretz gives really does come from leket, shich'chah and peah, or from the tithe for the poor, it is not subject to tithes and there is no concern of demai.
The wording of the Mishnah:
"Ne'emanin al haleket ve'al hashich'chah ve'al hapeah bish'atan" - poor amei ha'aretz are believed when they say that the produce they are selling comes from leket, shich'chah and peah that they gathered. "Bish'atan" - at the time when leket, shich'chah and peah are found in the fields; in the winter or the spring, when this is no longer relevant, there is no such credibility. In their season they are believed when they say that the produce they are selling is exempt from tithes.
"Ve'al ma'aser ani bechol shenato" - the tithe for the poor takes the place of second tithe in the third and the sixth year of the shemittah cycle: instead of bringing the tithe up to Jerusalem, one separates the tithe for the poor and gives it to a poor person. In a year of the tithe for the poor, the third or the sixth, a poor person is believed throughout that entire year when he says that what he is giving or selling comes from the tithe for the poor and is exempt from tithes.
"Uven Levi ne'eman le'olam" - a Levi receives the first tithe, and this is relevant in every single year, and therefore he is believed when he says that what he is giving comes from first tithe. True, the Levi too must separate a tenth and give the kohen terumas ma'aser, but just as the amei ha'aretz are not suspected of failing to separate terumah gedolah, so too a Levi who is an am ha'aretz is not suspected of failing to separate terumas ma'aser. He is believed that he separated terumas ma'aser, and once he says "this is from first tithe" - one may believe him, and there is no need to separate anything on account of demai.
"Ve'einan ne'emanin ela al davar shebnei adam nohagin ken" - all of their credibility to say that these items come from peah, from the tithe for the poor, from terumah or from first tithe applies only when what they are giving is of the type that people are accustomed to give as gifts to the poor.
And the detailed list of those items we will see in the next Mishnah.