Introduction to Tractate Demai
According to halakhah a person must separate terumah from his produce and give it to the priests. We will learn more about this in Tractate Terumah. Afterwards, he separates a tenth of what is left and gives that to the Levites. This is called maaser rishon or first tithe. The Levites take a tenth of the tithe given to them and they give it to the priests. This is called terumat maaser because it is terumah taken from the maaser. After the owner of the produce has taken out maaser rishon he must take out maaser sheni, or second tithe. This second tithe belongs to the owner and he must bring it to Jerusalem and eat it there (there will be a tractate about this as well). However, maaser sheni is only taken out in the first, second, fourth and fifth years of the sabbatical cycle. In the third and sixth years maaser ani, the poor mans tithe, takes the place of the maaser sheni. Before all of the terumah and tithes have been separated from the produce, the produce is called tevel and it may not be eaten even by priests.
The rabbis feared that amei haaretz, ignoramuses, did not separate tithes, although they assumed that they did separate terumah. Produce that comes from an am haaretz is called demai usually translated as doubtfully tithed produce. One who acquires demai must first separate from it maaser rishon, and from the maaser rishon he must take terumat maaser and give it to the priest. Note that the rules governing terumat maaser are strictonly priests can eat it and a non-priest who eats it is liable for death at the hand of Heaven.
Next he separates maaser sheni and brings it to Jerusalem. However, he does not have to give the maaser rishon to the Levite, nor does he have to give the maaser ani to the poor because it is only doubtful whether or not the am haaretz did not properly tithe his produce. Since we dont know whether the tithe actually belongs to the owner or to the Levite/poor person, the owner keeps it because he has current possession. The Levite or the poor person could take it from him only if he could prove that it was rightfully his.
Our tractate discusses the rules of Demai. One of the interesting things about the tractate is that it shows us a lot about how the rabbis, who did observe the laws of tithing, related to other Jews, Jews who were not so cautious about this mitzvah. Not trusting that someone had tithed their produce is somewhat akin to not trusting that someones home is kosher. It prevents you from eating together, and people who cant eat together, cant really socialize together.