The Tosafot is a medieval commentary on the Talmud. They take the form of critical and explanatory glosses, which are printed in almost all Talmud editions. The authors are known as בעלי התוספות
'Tosafot' literally means "Additions". Heinrich Graetz posits that the glosses are so-called as they are additions to Rashi's commentary on the Talmud. In fact, the period of the Tosafot began immediately after Rashi had written his commentary; the first tosafists were Rashi's sons-in-law and grandsons, it mentions Rashi's commentary very often.
Others, especially Isaac Hirsch Weiss, posit since many Tosafot, particularly Isaiah di Trani make no reference to Rashi, 'Tsafot' means 'Additions to the Talmud', that is to say, they are an extension and development of the Talmud.
Up to and including Rashi, the Talmudic commentators occupied themselves only with the plain meaning of the text which was mainly based on tradition which tangentially incorporated the reconciliation of seemingly contradictory texts, since it had not developed into a life leading text to the masses. After the beginning of the twelfth century as the Talmud become more popular and widespread the beginning of a real critical approach began. Thus some of Rashi's descendants like his sons-in-law and his grandson Samuel ben Meir (RaSHBaM), and desciples wrote commentaries on the Talmud after the manner of Rashi's, or wrote glosses to it in a style peculiar to themselves leading eventually to an entire movement which changed the way the Talmud was analyzed. They had in front of them certain brief notes or short paragraphs of the simple meaning and explanation of the text called 'Kuntros'. It seems like this was sort of a companion explanation that was commonly used and many times coincides with Rashi's commentary, which seems to be something that was very similar to Rashi's commentary and he may have based his commentary very much on these 'Kuntros'. It was so similar that it could even be characterized as Rashi plagiarizing this work, even though Rashi made wholesale changes.
The Tosafot do not constitute a continuous commentary, but rather deal with select passages of the Talmud but a thread can be seen when analyzing a string of paragraphs and back references which are common. The Tosafist movement spanned over 3 and half centuries, from the 11th to the beginning of the 14th. This 2 volume set explain who the authors of the commentaries were on each individual tractate and sometimes even two different authors are identified in a single Tractate but never three. Possibly because, originally, certain authors did not complete their work or parts were lost. The expertise of Urbach in figuring out these historical details is very impressive. He is the ultimate authority on this area of the Talmud and it is virtually impossible to reconcile different parts of the Tosafot commentary without referencing the essential information, found in these volumes, on the work and the movement. Another important benefit that these volume's afford, are the work and analysis of tracing the opinions of these Rabbi's in theirs or other's responsa literature which mentions their opinions, this is used to come to conclusions about the develpement of the normative Halacha at that time and of their lifestyle, culture, and how they spent their leisure hours etc.
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