Let's start at the beginning. The Oral Law was not meant to be written down, but for various historical reasons the material that was taught in the academies was later collected and edited into the text that we know today as the Mishnah and the Gemara. However, there was no such an extreme stringency with the text that we see for copying a Torah scroll. This resulted that there are minor differences in the various manuscripts.
In a later period the church began to learn Hebrew, and they started to criticise Jews for having certain parts regarding their religion in the Talmud that they considered derogatory and unacceptable. Therefore they initiated mass burnings of Talmud manuscripts mainly from the 13th century. With the advent of the printing press, the church required a written approval from the censor that they removed the sensitive parts.
Daniel Bomberg was famous for trying to minimise this intervention of the church, but still, his edition (published between 1519-1523) was not free of censorship. He also made great efforts to consult many different manuscripts in order to filter out scribal errors (Rashi had also done similar work in his comments). Although published in the 19th century, the famous Romm edition of Talmud from Vilna - the one that most 20th century editions used - was also censored. If we don't consider the censorship, this edition has very few errors, which was highly appreciated, when books were typeset manually. The history of these editions is discussed in detail in the book Printing the Talmud, kindly suggested by Double AA in comment.
Regarding the Jerusalem Talmud it is quite clear that Bomberg mainly used the Leiden Codex with three other manuscripts. For the Babylonian Talmud there were far more early manuscripts. The censorship mostly had an effect on aggadic material, where a certain person, called Yeshu was involved. After a while scribes tried to find euphemisms to avoid censorship and persecution. These cases are compared in Peter Schäfer's book, but a few examples are given also on the Wikipedia. Here you can find reference to many important and high quality manuscripts.
As the power of the church started to decrease in the 19th century, some publications started to list these omissions. There is a shorter one, called Chesronot haShas, published in Königsberg in 1860. However, there's a far more detailed work by Rabbi Raphael Rabbinovicz, who found in Munich an uncesored manuscript of Bavli from 1342. He compared this and other manuscripts with the printed editions and published the multi-volume Dikdukei Soferim, an extremely detailed and precise work, which may serve as a basis of similar analysis.
Fortunately the scope of the church and the halakhic decisors were quite different, so this censorship had a small effect on our religious laws. In newer editions these omissions were reinserted as well. Your example is a consequence of a scribal error, and in overall they should have an even smaller effect, as they are random. Personally, I only know one case (whether there should be 67 or 70 lines in Haazinu), when such and error affected halakhah, but it was in Mishneh Torah.
