The Midrash, in Judaism, is Jewish commentary on the Tanakh compiled in the 2nd century. In a Christian context, the New Testament is sometimes considered to be a midrash on the Old Testament.
The word "Midrash" comes from the Hebrew word darash, meaning "search" or even "commentary". It "entails searching the text for clarification beyond the obvious (Evans, TOTITN, p. 131). In other words, midrash is a method which involves commentary on a specific passage of the Bible. "In 'searching' the sacred text, the rabbis attempted to update scriptural teaching to make it relevant to new circumstances and issues. This was approach was felt to be legitimate because Scripture was understood as divine in character and therefore could yield many meanings and applications..." (Evans, JE, p. 381). One of the best examples in the NT is John 6:25-59 which comments on Exodus 16:4, Psalm 78:24 (cf. Jn 6:31). Jesus' words are considered by some scholars as a running "commentary" on this passage found in the book of Exodus.
One of the most significant midrashic methods is known as light and heavy. According to this method, if something is true in a less important, "light" situation, it proves important in a greater, "heavier", situation. For example, Jesus assures his disciples that since God cares for the birds (light), he also cares for them (heavy; Matt 6:26; cf. Luke 12:24). This method is usually recognized by an "if" statement followed by a "how much more...", found both in the words of Jesus and Paul (cf. Matt 7:11; Rom 5:10).
The rule of equivalence is seen when "passages clarify one another if they share common vocabulary (Evans, TOTITN, p. 132). Consider 1 Peter 2:4-8 which quotes Isaiah 28:16, Psalm 118:22, and Isaiah 8:14. Here, the term "stone" is used in equivalent regulation. Another example is Jesus' use of 1 Samuel 21:6 in Mark 2:23-28.