Midrash is both an interpretative activity and a genre of rabbinic literature that includes expansive elaborations on biblical texts, often for homiletic purposes.
Apocryphal literature, from the Greek term meaning “to be hidden,” refers to a vast quantity of noncanonical gospels, apocalypses, letters and other writings featuring biblical characters and events. (The term Apocrypha can also be used to refer to a collection of books that is considered canonical by Roman Catholics and Eastern Orthodox but not by Protestants and Jews.)
The pseudepigrapha, from the Greek for “falsely ascribed,” is a group of Jewish writings dating from the second century B.C.E. to the second century C.E. that did not make it into the canon. Among the texts are apocalypses, histories, psalms and books of wisdom that are falsely attributed to Adam, Moses and other biblical figures. For more on the pseudepigrapha, see David de Silva, “Why Did God Choose Abraham?” BR 16:03.