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When should you use a, an, or the before a noun? To answer that question, we need to get three things straight about the noun:

1. Are you using the noun to refer to something indefinite or definite?  For example, let's consider the noun "accessibility." You could talk about accessibility in an indefinite sense--"accessibility is important"--or you could refer to accessibility in a specific context--"the accessibility of this blog post is important."
2. Is the noun countable?  Literally, could you count it?   For example, water is not countable, but glasses of water are.  Similarly, economics is not countable but economies are.
3. If countable, is it singular or plural?  That is, if you could count units of the noun, how many are there: one or more?

Now, we're ready to work out which article to use.  Here are some general rules of thumb (though there are probably exceptions).

 

Use the if the noun is definite

• and countable, whether singular or plural 

The stapler is in that drawer.  Here, "stapler" is used in a definite sense: the writer is referring to a specific stapler; which one would be clear from the context. 

The papers on immigration reform are relevant. Here, "papers" is used in a definite sense: it is the papers on immigration reform.  

• and non-countable and singular

The relevance of paragraph 4 is questionable.  Here, "relevance" is used in a definite sense: it is the relevance of paragraph 4.  It is also non-countable and singular

Use a/an if the noun is indefinite AND countable AND singular.

A graph would be useful. Here, graph is indefinite, countable, and singular: it's not a specific graph, graphs can be counted, and there's only one mentioned.

Can you include an appendix? Here, appendix is indefinite, countable, and singular: it's not a specific appendix, appendices can be counted, and there's only one mentioned.

"A" is used before nouns that begin with a consonant sound (e.g., cat, Europe, and yodeler), and "an" is used before nouns that begin with a vowel sound (e.g., apartment, hour, oily).

Use no article if the noun is

• indefinite, countable, and plural

Papers are hard to write.  Here, papers is used in the indefinite sense: no specific paper is mentioned.  

Mistakes were made.  Again, mistakes is indefinite.  Classic government speak--it's not clear what the mistakes are.

• indefinite and non-countable.

Life is good. Here, life is used in the indefinite sense; used in this sense, "life" is not countable. 

OK, here's a quick quiz: the, a, an, or no article?

1. ........... US economy is recovering from ......recession.
2. ......... risk is at the heart of this model.
3. It was ......... useful exercise.
4. .......workers act in accordance with their identity.
5. I study ......... Spanish economy.
6. I am thinking about ........ big picture. 
7. I plan to write ....... paper on ........housing market.
8. .......... economy is ............ delicate thing.

(scroll down for answers)

1. The US economy is recovering from a recession.
2. Risk is at the heart of this model.
3. It was a useful exercise.
4. Workers act in accordance with their identity.
5. I study the Spanish economy.
6. I am thinking about the big picture. 
7. I plan to write a paper on the housing market.
8. An economy is a delicate thing.