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 once offered, seemed most 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.
It should be emphasized that the Brisker method is by no means 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 formula "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).
As an extreme example, Rabbi Chaim Soloveitchik observed that that the Talmud discusses the Sheva Brachot celebrations in the week following a Jewish wedding. The Talmud requires "panim chadashos" (meaning "a new presence" or "new faces"), i.e. a guest must be present at the Sheva Brachot celebration who had not attended the wedding. Elsewhere, the Talmud comments that once sacrificial meat has been burned to ashes, the ashes no longer have a sacrificial status, as "panim chadashos ba'u l'chan" -- "a new presence has arrived", meaning that the ashes are not the same as the meat. "So if you were at a Sheva Brachot party, and you looked around and everyone there had already been at this couple's wedding, why not just take some meat and burn it to ashes?", challenged Rabbi Chaim. Clearly, the phrase "panim chadashos" has different meanings in the contexts of wedding celebrations and sacrificial meat.
The famed yeshiva of Volozhin, arguably the first modern yeshiva, favored a traditionalist approach towards Talmudics under the leadership of the Netziv, which often required absorbing a great amount of Talmudic material to acquire a "general Talmudic feel" before analyzing a topic. Later, however, Rabbi Chaim Soloveitchik became a lecturer at Volozhin. At this point, around the year 1880, Rabbi Chaim's new methods first became public.
However, as Rabbi Joseph Soloveitchik suggested in his eulogy for the Brisker Rov, the full, true "Brisk approach" as we know it today was not developed until Rabbi Chaim Soloveitchik had been rabbi of Brisk for many years. The notes that Rabbi Chaim Soloveitchik used for his lectures at the yeshiva of Volozhin (years before he assumed the Brisk pulpit) are still extant today, and the approach found there is not as well-developed as in (his and others') later published works. The notes could best be described as "proto-Brisk lomdus", a term which could be used regarding the works of the Beis HaLevi as well. Several modern scholars agree with this notion of "proto-Brisk", and it can be heard in the lectures of Rabbi Dr. Aaron Rakefet-Rothkoff. Nonetheless, as seen above, even "proto-Brisk" was already different enough and popular enough to cause significant tension at the Volozhin yeshiva.
An additional major influence on the "Brisk approach" was a Rabbi Mendel Epstein of Slutzk. Rabbi Chaim "Brisker" Soloveitchik spent several early teenage years in Slutzk, where Rabbi Epstein served as his melamed (Judaics teacher for pre-college levels). Rabbi Chaim later claimed that much of the "Brisker derech" attributed to him was founded on Rabbi Epstein's approach; however, as a small town's melamed, Rabbi Epstein and his ideas never achieved fame. Thus, Rabbi Mendel Epstein's contribution to the "Brisker approach" might be compared to that of Sir Isaac Barrow, Isaac Newton's mentor, to calculus.
The Brisker method has a certain parallel in the Dor Revi'i (commentary on Hullin) of Rabbi Moshe Shmuel Glasner. Many scholars had been perplexed by the Rambam's rulings, as they had been used to understanding the Talmud according to the Franco-German school of Rashi and Tosafot, as opposed to the Babylonian Geonic school followed by Rambam. Rabbi Glasner insisted that Rambam's interpretations follow perfectly from the Talmud once he is interpreted on his own terms. Rabbi Glasner's methods coincided remarkably with those Rabbi Haim; Rabbi Glasner's methods caused a sensation in the Lithuanian yeshivot in the late 1920s and early 1930s yeshivot, producing astonishment that a Hungarian rabbi had independently formulated a method so similar to Rabbi Haim's.
When it first appeared, some scholars denounced the Brisk approach as "chemistry", as it sought to analyze each Talmudic law by breaking it down into components, whereas a traditionalist approach focused more on the entirety of the laws.
While the Brisker method has won acceptance in almost all yeshivas today, it has its opponents. These include Rabbi Avraham Yishayahu Karelitz (1878-1953) (known as the Chazon Ish), who felt that often the existing approach to a Talmudic portion was sufficient. Additionally, the Brisker method is not widely used in modern yeshivas which stem from the Mirrer Yeshiva (originally from Russia), which instead tend to stress single, unifying themes throughout Talmudic concepts, often focusing on only one Rishon if it is seen as the most "truthful" approach to a Talmudic passage. "Mir-style" yeshivas are thus seen generally as opposed to "Brisk-style" yeshivas, though there is very little personal animosity.
In Brisker yeshivas, the tractates studied deviate from the tractates popular in most yeshivas. Most yeshivas learn the Talmudic laws of money, property, marriage, and divorce. In Brisk, there is a greater tendency toward Kodashim tractates, as well as Nazir and Sotah (more ritually oriented) tractates in Nashim. Rabbi Joseph Soloveitchik is noted for a tendency to study tractates in Seder Moed, a tendency formalized by Yeshiva University's decision to learn a tractate from Seder Moed every four years.