Are really very simple, although for learners whose mother tongue does not use articles, they no doubt seem an unnecessary complication.
A is the indefinite article - a book, that is to say any book.
“I went to Whitcoulls yesterday to look for a book”. I did not care what book it was that I bought.
The, is the definite article, and is also a determiner, referring to a particular book.
“The book I bought was on sale for $5.00, and was well worth what I paid for it.”
An, is also the indefinite article, used before words that begin with a vowel or a vowel sound - which is why it is sometimes used before words that start with the letter 'h'. I was taught by my parents to say 'an hotel', because in former years, the 'h' in hotel was not pronounced. Now, for the most part, it is pronounced, and the tutor who accused me recently of linguistic snobbery for saying 'an hotel' was somewhat justified.
However, facts like this do not excuse the man I heard on the radio this morning who said 'a orange'. He had to insert a glottal stop to say it, but say it he did. Considering the pains I have taken to explain to students that they must say 'an orange', the pain caused to my English brain was considerable.
The, is also of course the definite article to use with countable plurals, for example: “The books in my bedroom are all over the floor”.
For uncountable plurals, we use 'some', 'any', many', 'a lot', for example:
“When you go to the supermarket, please buy me some milk” .
Although we can use 'a or the' when talking about uncountable nouns, if we refer to the container - for example: “the bottle of milk”, “the kilogram of rice” or 'a box of cocoa powder'.
There is a phenomenon over the centuries, where the letter 'n' has wandered back and forth between the article and the word, so that Shakespeare could say 'Marry, nuncle'. (Uncle, of course!)
Some words were altered centuries back.
For example, a newt was once an ewt (earlier euft and eft), a nickname was once an eke-name, where eke means "extra" (as in eke out meaning "add to"), and in the other direction, "a napron" became "an apron" and "a naddre" became "an adder."Source