Literally, an Arabic word meaning sayings. In a religious context it is the second primary text for Muslims (followers of Islam) after the Qur'an (or Koran). It consists of narrations of actions and sayings of Muhammad. These narrations are connected back to Muhammad by a chain of narrators (i.e. I heard it from so-and-so, who heard it from so-and-so ... who heard it from Muhammad) and there are strict rules concerning checks on the validity of hadith. They function as an elaboration and practical implementation of the principles explained in the Qur'an, since the Qur'an recommends looking to Muhammad's actions as a guide to behaviour. There are six main collections of hadith: Bukhari, Muslim, Abu Dawood, Nisa'i, Tirmizi and Ibn Majah.

For example, here is one of the most important hadiths, complete with its chain of narration (or at least one of them).

On the authority of Yahya b.1 Sa'id al-Ansari on the authority of Muhammad b. Ibrahim al-Taymi, on the authority of 'Alqama b. Abi Waqqas al-Laythi, who narrated it from 'Umar b. al-Khattab who narrated that the Prophet (peace be upon him) said:

Deeds are a result only of the intentions of the actor, and an individual is rewarded only according to that which he intends. Therefore, whosoever has emigrated for the sake of Allah and His messenger, then his emigration was for Allah and His messenger. Whosoever emigrated for the sake of worldly gain, or a woman whom he desires to marry, then his emigration is for the sake of that which moved him to emigrate."

Recorded in Bukhari and Muslim2.

1. "b" in this context is used as an abbreviation for "bin" or "son of"; as in Arab cultures generally, people are identified by their father (surnames were added later).
2. As mentioned above, Bukhari and Muslim are two large collections (nine volumes and six volumes respectively) of hadith. If a hadith is recorded in both, it is considered pretty much incontrovertible (in Arabic, it is described as "agreed upon" or "muttafaqun alayh").