An ELT Glossary : Definite, Indefinite and Zero Articles

a) Use of the articles

The definite article "the" introduces information which is shared between the writer/reader or speaker/listener. The indefinite article "a/an" or the “zero” article, on the other hand, introduce information which the writer/speaker assumes will be new to the reader/listener.

Look at this sentence from George Orwell: Winston Churchill, the former prime minister of Britain, once said...

Here, Orwell assumes you know who Churchill is - his role as Prime Minister is assumed to be shared, and therefore "the" is used. Compare it with:

Andrew Law, a former Prime Minister of Britain, once said that...

This time, I'm not assuming that you know who Law is - in fact I'm assuming that the information will be completely new to you. And so I use "a".

Another example would be:

I spent my holiday in Dax, a town near the south west coast of France.
 (I presume you've never heard of it)

I spent my holidays in Washington, the city on the east coast that is, not the state on the west coast. 
(I presume you already know of both).

To say Winston Churchill, a former Prime Minister.... would therefore sound as if you imagined the reader had never heard of him and needs to be given information on who he was. Which in this case would probably sound as if you were insulting the reader's general knowledge.

See these two articles bt Scott Thornbury for a fuller, and excellent,  explanation of the different ways in which information may be “shared” and therefore require the definite article : Articles 1 Articles 2

b) The indefinite and zero articles

We’ve already seen that the indefinite and zero articles are used when the speaker/writer presumes that the information is new to the listener/reader. However, which one is chosen depends on the grammar and phonology of the nouns chosen.

The indefinite article is used when the noun is being used as a singular, countable noun – ie it is singular in the context but can potentially be made plural (I need a cup of tea!) – and the zero article is used when the noun  is being used uncountably (I love cheese!) or in the plural (I love horses!). Notice that I said “used countably” rather than “is countable”. Although most nouns are generally used either countably or uncountably (so eg piano is usually countable – there were two pianos in the room -  and cheese is usually uncountable), any noun can be used either countably or uncountably if the context is right.  So eg, when types of cheese are being discussed then the noun is used countably : Cheshire, Cheddar and Stilton are three British cheeses.

And if you happened to consider a piano to be a food rather than a musical instrument, you would use it uncountably:
Baby termite: Mummy I’m hungry. What’s for dinner?
Mummy termite: Piano. Come into the living room and we’ll eat now.
Baby termite : Oh no! I hate piano!

With nouns used countably, the choice between the two forms of the indefinite article (a/an) depends on the first sound (NB sound, not letter) of the following noun – a is used before consonant sounds and an is used before vowel sounds. So :  A pen / A university but  An elephant / An umbrella

Want to know more?

Leech et al, An A-Z of English Grammar and Usage, Pearson Education Ltd.

(Amazon will often have both new and used copies of the books you want at prices much cheaper than those advertised - so don't be put off. Click on the link to check.  As I write there's one going for £9.76!)