Aggadah (Hebrew: אַגָּדָה ʾAggāḏā or הַגָּדָה Haggāḏā; Jewish Babylonian Aramaic: אַגָּדְתָא ʾAggāḏəṯāʾ; "tales, fairytale, lore") is the non-legalistic exegesis which appears in the classical rabbinic literature of Judaism, particularly the Talmud and Midrash. In general, Aggadah is a compendium of rabbinic texts that incorporates folklore, historical anecdotes, moral exhortations, and practical advice in various spheres, from business to medicine.
The Hebrew word haggadah (הַגָּדָה) is derived from the Hebrew root נגד, meaning "declare, make known, expound", also known from the common Hebrew verb להגיד.
The majority scholarly opinion is that the Hebrew word aggadah (אַגָּדָה) and corresponding Aramaic aggadta (אַגָּדְתָא) are variants of haggadah based on a common linguistic shift from haphalah to aphalah forms. However, a minority of scholars believe that these words derive from a separate Aramaic root נגד meaning "draw, pull, spread, stretch" (corresponding to the Hebrew root משך or נטה). 
According to the latter etymology, aggadah may be seen as "the part of the Torah which draws man towards its teachings", or the teachings which strengthen one's religious experience and spiritual connections, in addition to explaining texts. (See similar re Masorah – in the sense of "tradition" – at Masoretic Text § Etymology.)
The Aggadah is part of Judaism's Oral Torah, the traditions providing the authoritative interpretation of the Written Torah. In this context, the widely-held view in rabbinic literature is that the Aggadah is in fact a medium for the transmission of fundamental teachings (Homiletic Sayings—מאמרים לימודיים) or for explanations of verses in the Hebrew Bible (Exegetic Sayings—מאמרים ביאוריים). Rabbinic thought, therefore, understands much of the Aggadah as containing a hidden, allegorical dimension, in addition to its overt, literal sense. In general, where a literal interpretation contradicts rationality, the rabbis seek an allegorical explanation: "We are told to use our common sense to decide whether an aggada is to be taken literally or not" (Carmell, 2005).
Rabbi Moshe Chaim Luzzatto (1707-1746), discusses this two-tiered, literal-allegorical mode of transmission of the Aggadah in his well-known Discourse on the Haggadot. He explains that the Oral Law, in fact, comprises two components: the legal component (חלק המצוות), discussing the mitzvot and halakha; and "the secret" component (חלק הסודות), discussing the deeper teachings. The Aggadah, along with the Kabbalah, falls under the latter. The rabbis of the Mishnahic era (c. 10 to c. 220 CE) believed that it would be dangerous to record the deeper teachings in explicit, mishnah-like, medium. Rather, they would be conveyed in a "concealed mode" and via "paradoxes". (Due to their value, these teachings should not become accessible to those "of bad character"; and due to their depth they should not be made available to those "not schooled in the ways of analysis".) This mode of transmission nevertheless depended on consistent rules and principles such that those "equipped with the keys" would be able to unlock their meaning; to others they would appear as non-rational or fantastic.
In line with the above, Samuel ibn Naghrillah (993-1056), in his "Introduction to the Talmud", states that "Aggadah comprises any comment occurring in the Talmud on any topic which is not a commandment (i.e. which is not halachic) and one should derive from it only that which is reasonable." As regards this, Maimonides (1138-1204), in his preface to the tenth chapter of Tractate Sanhedrin (Perek Chelek), describes three possible approaches to the interpretation of the Aggadah:
Note that Maimonides' approach is also widely held amongst the non-rationalistic, mystical streams of Judaism—thus, for example, Rabbi Isaiah Horowitz, the Shlah HaKodosh (c. 1555-1630) holds that "none of these sometimes mind-boggling 'stories' are devoid of profound meaning; if anyone is devoid of understanding, it is the reader" (Shnei Luchos HaBris, introduction). See also the Maharal's approach.
The Aggadah is today recorded in the Midrash and the Talmud.
In the Midrash, the aggadic and halakhic material are compiled as two distinct collections:
Many of the Torah commentaries, as well as the Targumim, interpret the Torah text in the light of Aggadic statements, particularly those in the Midrash, and hence contain much material on Aggadah interpretation.
Throughout the Talmud, aggadic and halakhic material are interwoven—legal material comprises around 90%. (Tractate Avoth, which has no gemara, deals exclusively with non-halakhic material, though it is not regarded as aggadic in that it focuses largely on character development.) The Talmudic Aggadah, generally, convey the "deeper teachings"—though in concealed mode, as discussed. The aggadic material in the Babylonian Talmud is also presented separately in Ein Yaakov, a compilation of the Aggadah together with commentaries.
Well-known works interpreting the Aggadot in the Talmud include:
The Aggadah has been preserved in a series of different works, which, like all works of traditional literature, have come to their present form through previous collections and revisions. Their original forms existed long before they were reduced to writing.
The first traces of the midrashic exegesis are found in the Bible itself; while in the time of the Soferim the development of the Midrash Aggadah received a mighty impetus, and the foundations were laid for public services which were soon to offer the chief medium for the cultivation of Bible exegesis.
Much Aggadah, often mixed with foreign elements, is found in the Apocrypha, the Pseudepigrapha, the works of Josephus and Philo, and the remaining Judæo-Hellenistic literature; but aggadic exegesis reached its highest development in the great epoch of the Mishnaic-Talmudic period, between 100 and 550 CE.
The Aggadah of the Amoraim (sages of the Talmud) is the continuation of that of the Tannaim (sages of the Mishna). The final edition of the Mishnah, which was of such signal importance for the Halakah, is of less significance for the Aggadah, which, in form as well as in content, shows the same characteristics in both periods.
It is important to emphasize the fundamental difference in plan between the midrashim forming a running commentary (מאמרים ביאוריים) to the Scripture text, and the homiletic midrashim (מאמרים לימודיים). When the scholars undertook to edit, revise, and collect into individual midrashim the immense array of haggadot, they followed the method employed in the collections and revisions of the halakhot and the halakhic discussions. The form which suggested itself was to arrange in textual sequence the exegetical interpretations of the Biblical text as taught in the schools, or the occasional interpretations introduced into public discourses, etc., and which were in any way connected with Scripture. Since the work of the editor was often merely that of compilation, the existing midrashim show in many passages the character of the sources from which they were taken. This was the genesis of the midrashim which are in the nature of running haggadic commentaries to single books of the Bible, as Bereshit Rabbah, Eikah Rabbati, the midrashim to the other Megillot, etc. See Midrash for more details.
Ein Yaakov is a compilation of the aggadic material in the Babylonian Talmud together with commentary. It was compiled by Jacob ibn Habib and (after his death) by his son Levi ibn Habib, and was first published in Saloniki (Greece) in 1515. It was intended as a text of aggadah, that could be studied with "the same degree of seriousness as the Talmud itself".
Popularized anthologies did not appear until more recently—these often incorporate "aggadot" from outside of classical Rabbinic literature. The major works include: