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A Madh'hab (مذهب) is a school of Islamic law or fiqh (Islamic jurisprudence). Within Sunni Islam there are four mainstream schools of thought, which are accepted by one another, and the Shi'ite school of fiqh which (according to a fatwa by Al-Azhar, the most respected authority in Sunni Islam) is also now accepted by some Sunnis as a legitimate fifth school of Islamic Law. The five major schools of Islamic law agree on many things, including the death sentence for apostates.
The various schools of Islamic law all developed as theologians and jurists debated among themselves more than a hundred years after Muhammad's death on how to identify and interpret what Muhammad had left behind by way of oral traditions. Additionally, adherence to a school of Islamic law appears to be more a matter of geography than conscience, as followers of each school exist, for the most part, in certain geographical regions, often divided by country lines.
The nature of the schools of law
The schools of law differ primarily in the authentication and interpretive methodologies they employ to firstly determine which scriptures (hadiths, for the most part, since the Qur'an is considered authentic by consensus) are soundly attributable to Muhammad and to secondly determine how these scriptures should be understood and reconciled with one other in such a manner as to allow the derivation of legal rulings.
It should be noted that the schools of law are in themselves neither static nor homogeneous, as they have each developed and formalized a great deal since the times of their eponymous founders and consist internally of a diverse variety of opinions. As such, the schools of law are perhaps better characterized as enclosed arenas of debate wherein a common set of rules are followed. When it comes to inter-madh'hab debate, there is no common set of rules that is obliged beyond the thinly theological (such as the divinity of the Qur'an) and, as such, the topic of debate is frequently the rules themselves rather than the substantive rulings they give rise to.
Starting in the 18th century, Muhammad ibn abd al-Wahhab gave rise to what would become the modern day trend in Sunni Islamic thought known as Salafism. Salafism, almost necessarily wildly diverse internally, is the basic idea that the four classical schools of Islamic law have developed so cumbersome a set of legal methodologies that staying loyal to both the exact words of scripture and any respective madh'hab's interpretive methods is impossible. Lacking a strict method of interpretation, the Salafis look back to the practices of the early Muslims and Muhammad's companions to determine how scripture should be interpreted, and almost always prefer to just cite a hadith directly in response to a legal question rather than provide a systematically-derived, nuanced answer - though, since many hadith are in themselves unclear and apparently contradictory, this becomes a contentious and even impossible task, thus forcing either disagreement or a resort to some to a more rudimentary (and crucially less systematic or standardized) interpretive approach that brings about a reconciliation of the texts.
The five schools
The Hanafi madh'hab (Sunni)
The Hanafi madh'hab was founded by Imam Abu Hanifah al-Nu'man (d. 767) in Kufa, Iraq. The Hanafi madh'hab is adhered to in the Levant, Central Asia, Afghanistan, Pakistan, India, Bangladesh, the largest part of Egypt, Iraq, Turkey, the Balkans, and by large parts of the Muslim populations of Russia and China. Large religious movements that exist within the Hanafi sphere are the Barelvi and Deobandi movements. Altogether, it is estimated that Hanafis form a plurality of Muslims world wide (roughly 30% of all Muslims).
The Hanafi madh'hab is descended from the Ahl al-Rai' (the so-called "partisans of reason") of Iraq from the early Muslim community, of which Abu Hanifah was a part. As such, the interpretive methodology of the Hanafis can be broadly described as favoring reasoning over an uncritical regurgitation of scripture. In more concrete terms, this means allowing reasoning by analogy (qiyas) on legal/moral matters where scriptures are absent (the other madh'habs would come to adopt this concept to some degree, but the Hanafis adopted it first and arguably employ it most liberally). Another manifestation of this general preference for reasoning over, say, resorting to relying on weakly authenticated scripture (as the Hanbalis generally do), is the legal principle of Istihsan, or juristic preference. Istihsan is the practice of favoring an epistemologically and methodologically weaker opinion simply for the sake of "public interest" or maslaha.
As with all the other madh'habs, the views of the Hanafis school changed over time, are internally diverse, and in many instances disagree today with the opinions of its founder, Abu Hanifah. The two most important transmitters of the Hanafi school are Abu Hanifah's top two students, Abu Yusuf and Muhammad al-Shaybani.
The Maliki madh'hab (Sunni)
The Maliki madh'hab was founded by Imam Malik ibn Anas (d. 795) in the city of Medina, located in the Hijaz. The Maliki madh'hab is adhered to in North Africa, West Africa, the U.A.E., Kuwait, parts of Saudi Arabia, and parts of Egypt. The Murabitun World Movement is a religious movement existing within the Maliki madh'hab.
Unique to the Maliki madh'hab is its non-exclusive reliance upon Islamic scriptures and some degree of reason. Imam Malik, living in a Medina that had just a few generations ago hosted Muhammad and his companions, thought it was prudent to observe and emulate the religious practices and beliefs of his contemporaries, even if they could not marshal a hadith narration to justify themselves. This was because Imam Malik believed strongly in what some have dubbed the "living" tradition in addition to the "written" or "memorized" scriptural tradition. As a result, the Maliki madh'hab is the only school of Islamic law to rely o on what is known as 'Amalu ahl al-Madinah (or, "the actions of the people of Medina"). The Maliki madh'hab is also distinguished by its relative cosmopolitanism and leniency compared to competing schools of Islamic law (thus making it particularly popular among Western converts to Islam today, most famously Hamza Yusuf) due to Imam Malik having reportedly received questioners from all over the Islamic empire and thus needing to accommodate a wide variety of cultural and social milieus in his legal judgements.
It is said that of all of the four madh'habs, the Maliki and Hanafi madh'habs are most similar methodologically, as the Maliki madh'hab employs the principle of Istislah, also translatable as "public interest" (comparable to and in many ways indistinguishable from the Hanafi principles of Istihsan and Mashlaha). The primary difference is that whereas the Hanafi madh'hab prefers qiyas, or analogy, to istihsan, the Maliki madh'hab prefers istislah to qiyas. The Maliki madh'hab also prefers the practices of the people of Medina as well as the Ijma or consensus of Muhammad's companions over qiyas, but allows qiyas as well.
The most important book in the development of the formal Maliki madh'hab is Imam Malik's own collection of Hadith, 'Amal, and fiqh commentary called the Muwatta. The other key source in the Maliki tradition is the Mudawwana which was a compilation, for the most part, of the views of Imam Malik as compiled by his important students Ibn Qasim and Sahnun. In addition to the views of Imam Malik, the two compilers occasionally include their own reasoning (using Imam Malik's principles) on legal matters on which Imam Malik did not himself opine. As with the other madh'habs, the Maliki madh'hab is internally diverse.
The Shafi'i madh'hab (Sunni)
The Shafi'i madh'hab was founded by Imam Muhammad ibn Idris al-Shafi'i (d. 820), who, after being born in Gaza, would travel to Mecca, Medina, Yemen, Cairo, and Baghdad. The Shafi'i madh'hab is adhered to in parts of Saudi Arabia, parts of Egypt, Indonesia, Malaysia, Jordan, Palestine, the Philippines, Singapore, Somalia, Thailand, Yemen, Kurdistan, and parts of India.
The Shafi'i and Hanbali madh'habs, compared to the earlier Hanafi and Maliki madh'habs, generally disprefer and often disallow reasoning, qiyas, and public interest as a basis for legal rulings in favor of a more literal adherence to the words of scripture. Both the Shafi'i and Hanbali schools are, in this light, seen as descendants of the Ahl al-Hadith (lit. "partisans of hadith", translatable as "partisans of tradition") trend among early Muslims.
In Shafi'i methodology, blatant istislah or istihsan or anything that would amount to preferring a less methodologically or scripturally reliable opinion "merely" for the sake of "public interest" or "well-being" is prohibited, as it is seen as human interference in the derivation of a law code that must be strictly divine. While qiyas is allowed, it is relegated to fifth position and is preceded and often overwhelmed by the Qur'an, hadith, ijma' (scholarly consensus or consensus of Muhammad's companions), and the individual opinions of Muhammad's companions.
The foundational book in the development of the Shafi'i madh'hab is the Risala of Imam Shafi'i himself, wherein he outlines the methodology of jurisprudence in his madh'hab, as well as the specific rulings his methodology has led him to. The Risala is in many ways the first fiqh book to explicitly formulate an interpretive methodology in specific terms, and thus inspired and guided the later formalization of of the methodologies of the other three Sunni madh'hab's whose founders did not outline their own methodologies in such great detail during their own lifetimes.
The Hanbali madh'hab (Sunni)
The Hanbali madh'hab was founded by Imam Ahmad ibn Hanbal (d. 855), who, after growing up in Baghdad, would travel throughout Iraq, Syria, and the Arabian peninsula. The Hanbali madh'hab is adhered most commonly today in the form of its distant descendant known as Salafism, most prominently in Saudi Arabia. There are, however, still pockets of those who follow the more traditional and classical school of Hanbalism in parts of Saudi Arabia, Qatar, Syria, Oman, Yemen, and Iraq. It is the least popular of all the four schools of Sunni law today.
The Hanbali madh'hab is the most famously and strictly scripturalist tradition of all of the four Sunni madh'habs, and is thus considered the ancestor of the modern Salafi movement which seeks to eliminate interpretive methodology altogether in favor of directly referencing scripture. This is because the Hanbali madh'hab plainly disallows any employment of "public interest" (istislah or maslaha) or "juristic preference" (istihsan) in the derivation of rulings, always preferring authentic and clear scripture to relatively ambiguous or inauthentic scripture, even if the ruling is deemed harmful to human well-being. While qiyas, or reasoning by analogy, is allowed by later Hanbali scholars, it is heavily de-prioritized in favor of weak scripture and the individual opinions of Muhammad's companions (that is, Hanbali jurists prefer weak scripture over analogical reasoning, no matter how clear and straightforward this reasoning is, as well as companions' opinions). Indeed, Ibn Hanbal himself prohibited qiyas outright, and the principle was only adapted (taken from the Shafi'is) by later Hanbali scholars as a last-resort measure. Ibn Hanbal furthermore rejected the very possibility of ijma (consensus) among Islamic scholars beyond Muhammad's companions.
Of all the madh'habs, the Hanbali madh'hab at the latest time, almost 200 years after Muhammad's death, and was the most strict and uncompromising. It is also the Madh'hab that has changed most with time. In addition to allowing qiyas, modern Hanbali scholars allow some sort of limited istislah or istihsan as well.
In what can be described as a characteristic coincidence, the most famous work left behind by Imam Ahmad was his collection of hadith, the Musnad Imam Ahmad, rather than his works on legal methodology.
The Ja'fari madh'hab (Shi'ite)
The Shi'ite Ja'fari madh'hab was allegedly founded by Imam Ja'far al-Sadiq (d. 765), the 6th Shi'ite imam, who was born in Medina. The Ja'fari madh'hab is adhered to today in Iran, parts of Afghanistan, parts of Pakistan, parts of Iraq, and parts of Saudi Arabia. This madh'hab is the exclusive to Twelver Shi'ites in particular. Sunni scholars and critical historians hold that the actual attribution of this legal school to Imam Ja'far is unfounded.
The Ja'fari madh'hab relies far more heavily on Ijtihad (independent legal reasoning) rather than focusing almost entirely on the rigid methodological and scriptural constraints that characterize the four Sunni madh'habs. Beyond this generic description, the Ja'fari madh'hab is itself divided into two branches: the Usuli branch and the Akhbari branch. According to the Usuli (lit. "principle-based" or "method-based") branch, scholars of law can and should independently interpret scripture on the behalf on the "occulted" or "hidden" twelfth Shi'ite imam. This is the branch of Ja'fari fiqh endorsed by most modern Shi'ites and the state of Iran (and the ayatollahs). The Usuli branch also further distinguishes between what is described as "conventional fiqh" (which is supposed to be objective and unchanging) and "dynamic fiqh" (which is supposed to be heavily dependent on temporal realities). The other branch of Ja'fari fiqh is the Akhbari branch, which is today almost non-existent (except for a few "neo-Akhbaris" in India). This school of Ja'fari fiqh is far more rigid and methodological, and is more easily compared to the Sunni schools of law (even if the scriptures it references are different) than it is compared to the Usuli branch of Ja'fari fiqh.