X-bar theory is a theory of syntactic category formation. It embodies two independent claims: one, that phrases may contain intermediate constituents projected from a head X; and two, that this system of projected constituency may be common to more than one category (e.g., N, V, A, P, etc.).
The letter X is used to signify an arbitrary lexical category (part of speech); when analyzing a specific utterance, specific categories are assigned. Thus, the X may become an N for noun, a V for verb, an A for adjective, or a P for preposition.
The term X-bar is derived from the notation representing this structure. Certain structures are represented by X (an X with a bar over it). Because this may be difficult to typeset, this is often written as X′, using the prime symbol or with superscript numerals as exponents, e.g., X1. In English, however, this is still read as "X bar". The notation XP stands for X Phrase, and is at the equivalent level of X-bar-bar (X with a double overbar), written X″ or X2, usually read aloud as X double bar.
X-bar theory was first proposed by Noam Chomsky (1970), building on Zellig Harris's 1951 (ch. 6) approach to categories, and further developed by Ray Jackendoff (1977). X-bar theory was incorporated into both transformational and nontransformational theories of syntax, including GB, GPSG, LFG, and HPSG. Recent work in the Minimalist Program has largely abandoned X-bar schemata in favor of Bare Phrase Structure approaches.