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Hadith (الحديث) are texts, originally orally transmitted, relating to the words, sayings, and deeds of Muhammad. Their authority is usually considered second to that of the Qur'an. Islamic scholars generally consider Bukhari's collection to be the most reliable. 
Each hadith is composed of the isnad or authority on which the hadith stands (ex: "it is related by Irbad bin Sariyah that the Apostle said/did etc., etc.") and the matn or relation of the tradition itself.
Currently, the hadith are commonly used in addition to the Qur'an to determine the implementation of Sharia. The set of texts that comprise the hadith vary in each branch of Islam. Sunnis consult the hadith and decide Sharia through a process of analogy and consensus, whereas Shi'a depend on the "divinely inspired" judgment of the Imams. Ibadis have a seperate form of hadith.
Not all Muslims accept the authority of all hadith, and the very small Qur'an-only sect rejects them wholly. Not all hadiths are equal as well, many are considered weak by Islamic scholars but this also depends on the denomination.
The weakest are called Maudu. These hadiths have a tendency to be based on things external to Islam like superiority of a political ideology, race, town, or any sort of supremacist tripe. Many of these are outright lies used for personal gain that ignore basic history both within and outside of Islam. Unsurprisingly, many of the hadiths terrorists cite are Maudu.
Hadiths are commonly quote-mined by various sorts of bigots, both inside (Islamic extremists and authoritarians) and outside (Islamophobes) Islam. Islamic extremists and authoritarians tend to use quote-mined hadiths to justify their violent and hateful actions while Islamophobes quote-mine hadiths and use them as examples of how "evil" Islam is. Bonus points if they conflate the hadith with the Qur'an, even if there's a contradiction between the Qur'an and the hadith.