Lexically speaking, the word "hadeeth" [حديث], whose plural is ahaadeeth [أحاديث], is, "New, recent... existing newly, for the first time, not having been before... Information, a piece of information, intelligence, an announcement... a thing, or matter, that is talked of, told, or narrated..." 
In both the Qur'aan and hadeeth, the word has been used in reference to a religious communication, a story of a general nature, a historical story and a current story or conversation. 
As a technical term, a hadeeth is basically any report of the Messenger of Allaah's صلى الله عليه وسلم saying, action, tacit approval, manners, physical characteristic or biographical data. In other words, it is any report about the "sunnah," as defined by the scholars of hadeeth.
Every hadeeth is composed of two parts:
(a) Isnaad [إسناد], or chain of authorities, and
(b) Matn [متن], or the actual text of the hadeeth.
Both of these parts have to meet stringent requirements for the hadeeth to be accepted and considered true. In general, one can divide all ahaadeeth into five basic categories:
(a) Saheeh [صحيح] or authentic hadeeth;
(b) Hasan [حسن] or "good" hadeeth;
(c) Da'eef [ضعيف] or weak hadeeth;
(d) Da'eef Jiddan [ضعيف جداً] or very weak hadeeth, and
(e) Mawdoo' [موضوع] or fabricated, forged hadeeth. 
Actually, these can be broken down into two even more basic categories: accepted hadeeth (saheeh or hasan) and rejected hadeeth (da'eef, da'eef jiddan and mawdoo').
To be a source or authority of Islamic law, a hadeeth must be from the categories of saheeh or hasan. In order for a hadeeth to be saheeh or hasan on its own merit, it must meet the following five criteria:
(1) The chain, or isnaad, must be unbroken. In other words, each source must have received the hadeeth directly from the one on whose authority he is relating it all the way back to the Prophet. If there are any missing authorities, the chain would be considered broken and unacceptable.
(2) Every narrator in the chain must be of acceptable righteousness and character; in other words, each narrator must be morally fit. Impious people are not accepted for their impiety is a sign that they do not fear Allaah سبحانه و تعالى and, hence, they cannot be trusted to take extreme care in narrating the statements of the Prophet صلى الله عليه وسلم. If just one narrator in the chain does not meet this criterion, the hadeeth will have to be rejected.
(3) Moral characteristics are not sufficient. Each narrator must also be proficient and exact when it comes to narrating ahaadeeth. If a person is known to make lots of mistakes when narrating ahaadeeth, either from his memory or from his writings, his ahaadeeth will not be accepted.
(4) Both the chain and the text of the hadeeth must be such that they do not contradict what has been narrated through stronger means.
(5) Upon inspection of the different ways a hadeeth is narrated, it must be the case that no mistake or defect is spotted in either the chain or the text of the hadeeth.
If any of these conditions are not met, the hadeeth will be rejected as either weak (da'eef) or very weak (da'eef jiddan), depending on the magnitude of the weakness. Ahaadeeth which are graded da'eef or weak may be raised to the level of hasan if sufficient corroborating evidence is found for them. Ahaadeeth which are da'eef jiddan may never be raised because the nature of their weakness prevents them from being considered as supporting evidence or as being supported by other similar evidence. Of course, fabricated ahaadeeth are in a different category completely and would never, under any circumstances, be considered an authority in Islamic law.
The word Khabar [خبر] literally means "a report, news." It is used by some scholars of hadeeth as a synonym for hadeeth. However, others use the word hadeeth for what is attributed to the Prophet صلى الله عليه وسلم and what is attributed to other than the Prophet صلى الله عليه وسلم would be called khabar. Hence, a person specialising in the sunnah is called a muhaddith while someone concerned with history and other narrations is called ikhbaari. Imaam as-Suyooti رحمه الله notes that when the word "hadeeth" is used by itself, without any additional adjective decribing its source, it should only be used in reference to hadeeth of the Prophet (may Allaah exalt his mention and grant him peace). 
Athar [أثر] literally means the remnants or remains of something. Technically, it is used for what is narrated from the Prophet صلى الله عليه وسلم, his Companions رضي الله عنهم, their followers and other early scholars رحمة الله عليهم. A person who studies these reports and follows them is referred to as Athari. Imaam as-Suyooti رحمه الله says that athar should used only for what is narrated from the Companions رضي الله عنهم and the Followers رحمة الله عليهم and not for what comes from the Prophet صلى الله عليه وسلم, which should be termed hadeeth.  
 Lane Arabic-English Lexicon, vol. 1, p. 529
 Cf., Mustafa Muhammad Azami, Studies in Hadith Methodology and Literature (Indianapolis, IN: American Trust Publications, 1977), pp. 1-2.
 A fabricated or forged hadeeth is one which can be traced to an actual fabricator of hadeeth. When discussing hadeeth, many scholars do not even consider it a type of hadeeth.
 Cf., Muhammad Dhiyaa ar-Rahmaan al-Adhami, Mu'jam Mustalahaatul Hadeeth wa Lataa'iful Asaaneed (Riyadh: Adhwaa as-Salaf, 1999), p. 148; Muhammad al-Manshaawi, Qaamoos Mustalahaatul Hadeeth an-Nabawi (Cairo: Daar al-Fadheelah, n.d.), p. 56.
 Cf., Muhammad Dhiyaa ar-Rahmaan al-Adhami, Mu'jam Mustalahaatul Hadeeth wa Lataa'iful Asaaneed (Riyadh: Adhwaa as-Salaf, 1999), p. 8; Muhammad al-Manshaawi, Qaamoos Mustalahaatul Hadeeth an-Nabawi (Cairo: Daar al-Fadheelah, n.d.), p. 16.
 The Authority and Importance of the Sunnah, pp. 26-28.