The following short list is given only as an introduction to the terms. You should rely on more substantial reference works for your understanding of these terms. My purpose here is simply to sensitize you to their importance for understanding how the New Testament and Old Testament mesh.
Aramaic -The language most common to first-century Jews in and around Jerusalem and Judea.
Greek – the language most common to all peoples living in the Mediterranean world of the first century.
Midrash (singular), Midrashim (plural) – This definition taken entirely from Donald Juel’s Messianic Exegesis: “The most frequently used term to describe biblical exegesis in the postbiblical era is “midrash”-from the Hebrew darash, “to seek,” “to interpret.” The term is used in several different ways. It refers first to a body of literature, the midrashim, scriptural commentaries or collections of individual reflections on biblical material which began to appear in the fifth century. Second, the term is used to speak of a particular comment on a biblical phrase or verse-a midrash. Finally, the word is used more loosely to speak of the approach to scriptural interpretation characteristic of”midrashic” literature.” Later, Juel adds “Midrash was the vehicle by which meaning was actualized in the present.” [Note: “postbiblical” in this context should be understood as “post-Old Testament; note also that Juel spends p. 35-59 of his book on a section titled “Midrashic Exegesis,” so it is not an easy concept to encapsulate.]
Pesher (singular), Pesherim (plural) – a form of midrash particular to the Qumran community (of the Dead Sea Scrolls) offering, unlike rabbinic midrash, but a single interpretation of the text. There is “continuous pesherim” which offers commentary on each succeeding verse of a given biblical book (e.g. Habakkuk) or unit (e.g. as psalm), and “thematic pesherim” which pulls together texts from various locations in Scripture when dealing with a common theme. Donald Juel points out that pesher (given that it existed in a sect) was presented as revelation whereas rabbinic midrash (which allowed a variety of views) was presented as reason. However, Juel also says, “It is possible to overstate the differences between pesherim and midrashim, however, or at least to misunderstand the differences.”
Septuagint – The Old Testament translated from Hebrew to Greek; often referred to by the LXX (an allusion to the 70 original translators).
Targum (singular), Targumim (plural) – Translation, paraphrases, or interpretation of the biblical text into Aramaic (or perhaps Greek); necessary since Hebrew had fallen out of common use. This practice is traced to the time of Ezra. He also says, “In many cases, the Targumim are fairly literal translations. Often, however, the translations-or paraphrases-presuppose creative interpretations of the Hebrew known from other exegetical literature…[and the] Targumim are important as evidence of interpretive traditions…”