The correct use of the article depends on the kind of noun it goes with. We'll use the numbers 1-3 to identify the kind of noun as follows:

1-	countable singular		the / a
2-	countable plural		the / x / some
3-	uncountable		the / x / some

 
In the examples below, the numbers identify one of those 3 cases.

When talking in general there is no article
2- I like apples   
2- tigers live in Asia    
3- grass is green and water is transparent

Note: French wine is very expensive (we are not talking about all the wine in general, but we are talking about all the "French wine" in general, so we use no article)

When talking about something particular and known, we use "the" in every case
1- the president of Uganda is ill    (we know what president we are talking about)
2- the elephants of Africa are bigger than those in Asia
3- the wine you sell is excellent

When talking about something particular but unspecified, we use the indefinite article:
1- I want an apple  (I don't want a specific apple, any apple is fine, this or that or that one there; if it's an apple, it's ok)
2- I want apples = I want some apples
3- I want wine = I want some wine
 
The indefinite article "a" ("an" before a vowel) is singular, and so, we cannot use it for plurals or uncountables. Nevertheless, in those cases we can normally use the indefinite article "some" ("any" for interrogatives and negatives).

- I have a book
- I have books = I have some books
- I don't have books = I don't have any books
- Do you have books? = Do you have any books?

In these cases, the version with Some/Any is the most common one.
 
REMEMBER! "some/any" is an indefinite article, and so, it can only be used when the quantity or number is unspecified and not very big, otherwise, we do not use it (if you have thousands of books you cannot say "I have some books", you have to say "I have books / I have many books"). It means something similar to "a few / a little".

EXAMPLES:
 
What is there in the box?
 
- there is a pen                (just one)
- there are some pens     (a few pens, more than one but not many)
- there are three pens     (not *some because it is a definite number )
 
If there are 5000 pens...
- there are pens
- there are a lot of / lots of pens    (not *some because the number is big)
- there are many pens           
 
You can’t say:  *you’ve got some beautiful legs   
because even if  the number of legs is not big, it is certainly not an indefinite number, we all know that you have two!!

  NOTE FOR SPANISH SPEAKERS

The use of articles in Spanish is quite similar to that in English, but there are particularly two cases where Spanish people have many problems:

When talking in general there is no article (but in Spanish you use the definite article)
- I like apples     = me gustan las manzanas
- tigers live in Asia     = los tigres viven en Asia
- grass is green and water is transparent      = la hierba es verde y el agua transparente

You cannot use singular countable nouns alone (without: a / the / my, etc)
- un hombre sin cabeza = a man without a head
- nunca lleva abrigo = he never wears a coat
- no tengo coche = I haven't got a car
- tengo novia = I have a girlfriend
- busco trabajo = I'm looking for a job
 
Professions always use "a" (in singular, of course) he's a teacher - they are teachers