Answer:
   
   Medrash is the material associated with the text of the Torah that
   isn't in the text itself. These can be halachic thoughts or aggadic
   (non-halachah) ones. The medrashei halachah predate the mishnah, and
   were the original attempt to record the Oral Torah, and come from the
   generations right before the Mishnah. There were two schools of
   medrashei halachah, Rabbi Yishma'el's and Rabbi Akiva's. They had
   different basic assumptions about the text, and different means of
   associating rulings to the text.
   
   Medrashei Aggadah are generally later recordings, such as Medrash
   Rabba (the greater medrash; volumes are named "Bereishis Rabba", "Ruth
   Rabba", and the like) or Yalqut Shim'oni (Simeon's Selections).
   However, the material in the compilation dates back to the mishnaic
   era. These are sometimes similar in form to medrashei halachah,
   deriving things hermeneutically from the text. They are also sometimes
   written in the form of parables. This is so as to balance the need of
   recording the Oral Torah during traumatic shifts of exile with the
   need of keeping the Oral Torah oral. Because so much of medrash is in
   the form of parables, people loosely use the word "medrash" to refer
   to aggadic (non-halachic, ie philosophical and ethical) material from
   the Talmud(s) told in this manner.
   
   (Note: "Midrash" is proper Aramaic grammar. "Medrash" is proper
   Yiddish and yeshiva jargon.)