The Brisker method, or Brisker derech, is a reductionistic approach to Talmud study innovated by Rabbi Chaim Soloveitchik of Brisk, as opposed to the traditional approach which was rather holistic. It has since become popular and spread to yeshivas around the world. The Brisker method is also known as the "conceptual" approach to Talmud study, and is often referred to simply as lomdus (lit. "analytical study").
Broadly speaking, before the Brisker method, Talmudic texts were taken at "face value" unless there was a compelling reason not to. If a contradiction between two texts was discovered, then it became necessary to reinterpret one or both texts in order to reconcile them. But there was no standard method by which to perform this reconciliation. Any explanation which one offered, which seemed reasonable, would be accepted.
The Brisker method replaces this approach with a methodical search for precise definitions of each concept involved in the discussion. Once the mechanism by which a law works is rigidly and correctly defined, it can become clear that one aspect of the definition applies in one situation but not another. Therefore, the final halacha will differ in the two situations, even if they superficially appear to be very similar.
Often an entire series of disagreements among the Rishonim (Talmudic commentaries from roughly the period 1000–1500) may stem back to a subtle difference in how these Rishonim understand a line from the Talmud. The Brisker method can provide a precise formulation of how each Rishon understood the topic, and thus account for their differences in opinion. This approach is most spectacular when a whole series of debates between two Rishonim can be shown to revolve around a single "chakira", or difference in the understanding of a Talmudic concept.
The Brisker method is not a total break from the past. Rabbis before Brisk sometimes made "conceptual" distinctions, and Brisker rabbis can still resolve issues without recourse to the terminology they invented. The difference is one of focus and degree. Non-Brisk analysis tends to formulate "conceptual" definitions only when necessary, while for Briskers, these definitions are the first and most common tool to be used when approaching a Talmudic issue.
One example of Rabbi Chaim's emphasis on the value of precise definition can be found in quote: "One approach which answers three different problems is better than three different approaches to individually solve the three problems" (a corollary of Occam's razor).