In linguistics, the word zero can act as a modifier for words and parts of speech. It generally means that there should be an x, but there isn't, usually as a characteristic of a specific dialect. 'Zero definite article' refers to the lack of a definite article; in the English language this means 'the'. A noun appearing without either a indefinite or a definite article may also be called a bare nominal.
English has a lot of dropped definite articles (although not as many as most Romance languages). For example we say "I'm going to school", not "I'm going to the school". Some of these, such as 'prison', 'church' and 'bed' are dropped world over, to the point that using the definite article would sound odd ("I'm going to the church" or worse, "I'm going to the bed"). Musical instruments are common victims of the zero definite article effect ("I play piano", "Bob is on drums").
Other zero definite articles are specific to a dialect; some UK dialects will allow the speaker to drop the 'the' before words like 'store' and 'road'. It's interesting to note that in most of the English speaking world it is apparently common to say "he's in hospital" -- something that an American English speaker would never do. On the other hand, an American is more likely to say "I saw him on ` television" than are English English speakers.
This is nothing new; you can find dropped articles in Shakespeare ("All hail, Macbeth, thou shalt be ` king hereafter!"). Nor are they the leftovers of a bygone era; 'on television', 'off record' and 'off camera' are clearly new constructions. As far as I know, no one has come up with a good theory as to why some phrases drop articles and others do not.