An Arabic word meaning literally "Divine Saying". In the context of the religion of Islam, it means something that God (or Allah, as it is said in Arabic) said to a Muhammad, but was not meant to be part of the Qur'an. It may be in the form of a story about "Allah". For example, here is one well-known Hadith Qudsi:

On the authority of Abu Hurayrah , who said that the Prophet of Allah (peace be upon him) said:

When Allah decreed the Creation He pledged Himself by writing in His book which is laid down with Him: My mercy prevails over my wrath.
Abu Hurayrah is the narrator of the incident; this is only the last link in the chain of narration. There is usually a complete list of names of who said it until it was written down. For brevity, this has been omitted. The Prophet tells of an instance where God told him something.