Jews study the Tanakh (Bible) on multiple levels:
The first level is פְּשָׁט peshat, taking the text at face value, in context . This doesn’t quite mean “literal”, because we of course take into account idioms, metaphors, personification, etc. The peshat is the message that the original author intended to get across to the original audience.
The second level is the distinctively Jewish way of reading our Bible: דְּרַשׁ ,derash. This the way that Ḥazal (חז”ל) – the rabbis of the Mishnah, Midrash and Talmuds – interpreted the text. In derash we ask why the text is phrased the way that it is. Rabbinical literary techniques plumb the depths of the text to find new meaning, or may bring out lessons that may not have been intended by the original authors.
Discussions by Ḥazal (חז”ל) reveal that, in some cases, they felt that derash was discovering the original meaning of the text, while in other discussions they clearly understood derash as filling-in-the-blanks – creating new meaning. Often they were writing Biblical homilies. See Are Midrash literally and historically true?
During the medieval era both schools of thought continued: Some meforshim (classical Bible commentators) such as Rashi, often accepted much derash as literally and historically true, while others (Rashbam, Abraham Ibn Ezra) felt otherwise.
Conflating the derash with the peshat later became a defining characteristic of fundamentalist versions of rabbinic Judaism. Understanding that they are not identical became characteristic of non-fundamentalist versions of rabbinic Judaism.
Ari Marcelo Solon writes “Rashi’s grandson, Rashbam (R. Samuel ben Meir) clearly
distinguished between peshat and derash. His terminology relating to the peshat category is well-defined. Rashbam consistently interpreted in accordance with the peshat method; that is to say, he limited himself to the text itself, interpreting it according to its vocabulary, syntax and context, in relation to biblical parallels, according to common sense as well as derekh eretz (what is customary). Unlike Rashi, Rashbam did
not integrate biblical text and Midrash. It was Rashi who paved the way towards a clear distinction between peshat and derash in the writings of his successors. Yet in his commentaries, such a distinction still remains unrevealed.”
Haredim (ultra-Orthodox) may deny that there is any difference between peshat and derash. They characterically teach that we are obligated to accept the derash as if it is the literal, original and only interpretation of the Bible. They may refer to any other approach as heretical.
In contrast, rabbis who appreciate great medieval Bible commentators such as Abraham Ibn Ezra and Rashbam, or who follow philosophical rationalism, have the opposite approach: Such rabbis are found within some of Modern Orthodoxy and all of non-Orthodox Judaism (Conservative Judaism, Masorti, Reform, etc.)
To see examples of Jewish bible study, see Jewish Tanakh (Bible) commentaries in English.
Modern Orthodox Rabbi Shalom Carmy (Yeshiva University) explains the difference between peshat and derash like this
1. Peshat–what text meant for first generation audience. Derash- what it may mean in retrospect. (Rabbi D.Z. Hoffmann says this).
2. Peshat– what’s in the lines; Derash- what’s hinted at between the lines, OR
2′. Peshat–what’s in the text; Drash- “filling in gaps” of what’s not explicit in text.
The relations between these levels is complicated & function differently in Halakhic and narrative contexts.
There are also ambiguities–what’s written in the first chapter of a book often has one meaning when you read the book the first time and another meaning when you get to the end. Likewise what a pasuk means in Shemot may appear different after you have reached Dvarim.
Correctly Construing Biblical Verses Upon which Halakhot Claim to be Based, Professor Rabbi Marty Lockshin
Ibn Ezra vs. Rashbam – Can The Torah Contradict Halacha (Jewish Law)?
“Peshat and Derash: Plain and Applied Meaning in Rabbinic Exegesis” by Rav Prof. David Weiss Halivni (Oxford U. Press 1991)
Book: Rabbi Abraham Ibn Ezra: Studies in the Writings of a Twelfth-Century Jewish Polymath, Edited by Isadore Twersky and Jay M. Harris. Chapter Abraham Ibn Ezra as an Exegete, by Nahum M. Sarna
What do we do when a verse in the Torah says one thing but halakha, Jewish law, attributes a very different meaning to it? Some people engage in fundamentalist wordplay to conclude that there’s no difference between the peshat of the Torah, and Halakhah. But such differences exist; Even the Talmud notes this:
In the nineteenth century, Samuel David Luzzatto (Shadal; 1800-1865) developed a new way of solving the peshat-halakha dilemma, suggesting that midrash halakha (rabbinic interpretation of biblical legal texts) often represents rabbinic legislation, and NOT biblical commentary. He makes his clearest and most detailed statement on the topic in his commentary on Parashat Tzav…. Shadal’s approach to the peshat–derash issue is novel and simple: Whenever the peshat says one thing and the midrash says something very different, Shadal says that the peshat is what the Torah means and the midrash represents rabbinic legislation, not biblical interpretation…. From a halachic point of view, this approach may be problematic: these laws that were connected to biblical verses by means of a derashah were standardly considered by the rabbis to be of Torah, not rabbinic, origin (דאורייתא, not דרבנן), as Shadal’s approach apparently implies. Remarkably, for Shadal, the classical rabbis were religious reformers who changed the laws of the Torah, making them less stringent. Shadal lived in the early days of Reform Judaism and took issue with its innovations. Accordingly, he takes pains to distinguish the motivations of the classical rabbis from what he understood to be the motivations of his more liberal contemporaries [Classic German Reform Judaism]