Short answers, pulled from the story.
Fiqh is the Arabic term for Islamic jurisprudence, and the word literally means deep understanding or full comprehension. It refers to the human understanding, research, and practice of sharia, the divine law revealed in the Quran and the sunnah.
Sharia is the divine Islamic law revealed in the Quran and the sunnah, which Muslims consider immutable and infallible. Fiqh is the human effort to interpret and develop that law through ijtihad, and it is considered fallible and changeable.
The four roots of law are the Quran, the sunnah, ijma, and qiyas. They were codified by Muhammad ibn Idris ash-Shafi'i, who lived from 767 to 820, in his book ar-Risalah.
Sunni practice has four prominent schools: Hanafi, Maliki, Shafi'i, and Hanbali, each named for the classical jurist who taught it. Shia schools are based on the Ja'fari school and include the Twelver, Isma'ili, and Zaydi, while the separate Khawarij tradition produced the Ibadi school in Oman.
A faqih is a person trained in fiqh, with the plural fuqaha. An individual who exercises ijtihad, the independent reasoning used to derive rulings, is called a mujtahid.
Islamic legal institutions developed by Muslim jurists during the Islamic Golden Age resemble later Western ones. The Waqf parallels English trust law, the Hawala influenced the agency of common law and the French aval, and the scholar John Makdisi linked the Islamic lafif of twelve witnesses to English jury trials under Henry II.