The Gemara is the part of the Talmud that contains rabbinical commentaries and analysis of the Mishnah. After the Mishnah was published by Rabbi Yehuda HaNassi, the work was studied exhaustively by generation after generation of rabbis in Babylonia and the Land of Israel. Their discussions were written down in a series of books that became the Gemara, which when combined with the Mishnah constituted the Talmud. The rabbis of the Gemara are referred to as Amoraim. Unlike the Mishnah, the Gemara is mostly written in Aramaic, the Jerusalem Gemara in Western Aramaic and the Babylonian in Eastern Aramaic, but both contain portions in Hebrew.
The analysis of the Amoraim is generally focused on clarifying the positions, words and views of the Tannaim. These debates and exchanges form the "building-blocks" of the gemara; the name for a passage of gemara is a sugya (סוגיא; plural sugyot). A sugya will typically comprise a detailed proof-based elaboration of the Mishna. Every aspect of the Mishnaic text is treated as a subject of close investigation. This analysis is aimed at an exhaustive understanding of the Mishna's full meaning.
In the Talmud, a sugya is presented as a series of responsive hypotheses and questions - with the Talmudic text as a record of each step in the process of reasoning and derivation. The Gemara thus takes the form of a dialectical exchange. (By contrast, the Mishnah states concluded legal opinions - and often differences in opinion between the Tannaim. There is little dialogue.) The disputants here are termed the makshan (questioner, "one who raises a difficulty") and tartzan (answerer, "one who puts straight").
The gemara records the semantic disagreements between Tannaim and Amoraim. Some of these debates were actually conducted by the Amoraim, though many of them are hypothetically reconstructed by the Talmud's redactors. (Often imputing a view to an earlier authority as to how he may have answered a question: "This is what Rabbi X could have argued..."). The discussions generally go like this:
We learned in a Baraisa/Mishna OR Rabbi 'A' said XYZ.
Rabbi 'B' asked, but we learned a something else that contradicts that in 'X' place.
Rabbi 'C' answers that here it is talking about one thing and here another OR Rabbi 'C' brings a proof to one opinion.
Sometimes, an argument can take more than one page to be resolved. Though the gemara generally does not say who is correct, there is a commentary that usually decides which is correct. Additionally, sometimes the gemara can not find an answer, and declares that there is no answer, the most usual expression of which is "Taiku," "Leave it [for the Messiah to answer]."