Answer:
   
   The Mishna is basic compilation of the Oral Law, and was written down
   around 200 CE. However there is another compilation of Oral Law from
   that time period--the Tosefta. Rashi (in his commentary on BT
   Sanhedrin 33a) writes that the Mishna was redacted by Rabbi Judah
   Ha-Nasi in consultation with members of the Academy, while the Tosefta
   was edited by Rabbis Hiyya and Oshaiah on their own. This gave the
   Tosefta less authority than the Mishna; today, the Tosefta is treated
   a supplement to the Mishna.
   
   The word 'tosefta' means 'supplement'. The Tosefta is a Halakhic work
   which corresponds in structure almost exactly to the Mishna, with the
   same divisions for sedarim (orders) and masekhot (tractates). It is
   mainly written is Mishnaic Hebrew, with a few Aramaic sentences. The
   actual writing is called the Tosefot or Tosefos, depending on your
   Hebrew dialect.
   
   Tosefot was produced by a school of French Rabbis of the 12th century.
   Their thoughts were combined into a commentary on the Babylonian
   Talmud.
   
   Tosefot is found on the outside of each page (on the left of the left
   page, or the right part of the right one) wrapped around the text.
   Rashi, who was father and grandfather of a number of the Tosafists
   appears on the inside, nearer the binding.
   
   The thrust of the commentary is to resolve the meaning of the page
   both when internally difficult and they were dissatisfied with Rashi's
   understanding, or when there are difficulties understanding the text
   in light of what is written elsewhere in the Talmud. (Rashi doesn't
   directly refer to the latter kind of problem.)
   
   Professors Agus and Ta-Shma argue that Tosafot set out to explain
   Ashkenazic practice in light of the Talmud. The Babylonian Talmud had
   gained exceptance as /the/ snapshot of the Oral Law. However,
   Ashkenazic rulings and customs had until then been justified based on
   other sources as well, the halachic medrashic texts, the Jerusalem
   Talmud, etc... This is because of the number of Ashkenazic Jews who
   came from Israel (via Italy), not Babylon. Now that the Babylonian
   Talmud gained prominance, addressing questions of how ideas found in
   Ashkenaz fit that greater picture became more urgent. They do not
   overtly refer to this mission, but many of their answers do end up
   providing such explanations.
   
   Also, at the time, the Tosefists were one of two schools of thought.
   There were also the Chassidei Ashkenaz, who were a pietist movement
   that had a greater focus on going beyond the letter of the law. There
   was much friction between the Tosefits and the Chassidim, much like
   what happened with the current Chassidic movement, when it was founded
   in the late 18th century.