Definition of "hadith:"

Hadiths are regarded as a narration on the Sunnah (lived example) of Muhammad.  It includes reported sayings, actions, and traditions of Mohammad and his companions.
M.M. Azami formally defines "hadith" as follows:

"According to Muhaddithiin [scholars of hadith] it stands for 'what was transmitted on the authority of the Prophet, his deeds, sayings, tacit approval, or description of his sifaat (features) meaning his physical appearance. However, physical appearance of the Prophet is not included in the definition used by the jurists.' "
"Thus hadith literature means the literature which consists of the narrations of the life of the Prophet and the things approved by him. However, the term was used sometimes in much broader sense to cover the narrations about the Companions [of the Prophet] and Successors [to the Companions] as well." ₁

The Qur'an says: "And whatever the Messenger gives you, take it, and whatever he forbids you, leave it. And fear Allah: truly Allah is severe in punishment." ₂ The Messenger refers here to the Prophet Muhammad. These writings are not regarded as having the same status as the Holy Qur'an, which is considered to be God's word.
The accuracy of Muhammad's sayings was confirmed by his contemporaries -- generally his companions; i.e. his immediate followers. Some Muslims regard all of the the Hadiths as being valid. Some historians question the accuracy of some passages. For example, Historian Bernard Lewis commented on a saying attributed to the prophet that some scholars believe is invalid:

"If anyone insults me, then any Muslim who hears this must kill him immediately."

Some terrorists partly believe that, based on this Hadith, all Muslims have a duty to kill Americans when they have the opportunity.
The great Islamic scholar Yahya bin Sharaf Ul-Deen An-Nawawi compiled a collection of 43 of the most important sayings of Prophet Muhammad (PBUH). It is is now known as "An-Nawawi's Forty Hadiths" ₃