3am at night, Tanya suddenly asked me this question.  

Curiosity kills a cat, but not a young scholar.  Good question although it is a little difficult to fully explain X-bar theory (or any other Chomsky's theory) at high-school level, but you are special, a linguist by birth (kidding), so let me give it a try.  Some illustration will give you some idea of what it is and what Chomsky intends to do with this theory.

Chomsky is obsessed with mathematical abstraction of linguistic theory and he tries to build his theory that is most general and ultimately uncovers the language universals underlying all human languages.  For this goal, he has this X-bar theory.

Basically, language has different levels of tokens (or units), let us take the nouns as a simple example: (1) basic unit is word (e.g. girl); (2) the next level is group (e.g. beautiful girl); (3) the next next level is phrase (e.g. this beautiful girl).  

Chomsky labels word-level as bar 0; group-level as bar 1; and phrase level as bar 2.  

X refers to noun, verb, or adjective.  

So X in terms of nouns is a Noun (N); X-bar is therefore a Noun Group (NG); X-double-bar is Noun Phrase (NP).

Sounds simple, er? a noun phrase can be diagrammed as follows:

Also see the trees in my following blog for NP (Noun Phrase) "John's car wash" and VP (Verb Phrase, some call it a sentence S, depending on different theorization of syntax) "John washed the car":

This shows that verb tree looks similar to noun tree.  That is exactly what Chomsky wants to achieve: different types of phrases share the same type of structure in diagramming.  By doing this, he goes one step further to abstraction and common universals.  

Chomsky is a very intelligent philosopher and mathematician in addition to a giant of linguistics.  (He is also THE most quoted man in the world and arguably the best known "political dissident" in America.)

Hope it does not confuse you or make you bored.

My bog in Chinese on X-bar theory:

http://blog.sciencenet.cn/blog-362400-698434.html

English wiki for X-bar theory:

http://en.wikipedia.org/wiki/X-bar_theory