1) The noun has been previously mentioned.

I saw a funny-looking dog yesterday [first mention, indefinite]. It looked like a cross between a Pekingese and a German Shepherd. When it saw my cat, the German Shepard ran away [second mention, definite].

2) A superlative or ranking adjective makes the noun's identity specific.

• The tallest girl in the class is 6'2" tall. [There can be only one girl who is the tallest.]
  
• Please read the 4th paragraph on page 3. [There can be only one 4th paragraph.]
  
• Today is the most important day of my life. [There can be only one day that is the most important.]

3) The noun describes a unique person, place, or thing.

The earth revolves around the sun once every 365 days. [There is only one earth and only one sun--in our solar system, that is!]

4) A modifying word, phrase, or clause follows the noun and makes it clear which specific person or thing you are referring to. But not every noun that is modified in this way is definite; it depends largely on the situation and on what you can reasonably expect your listener/reader to know about.

"Do you remember the girl who went camping with us?"
[Using "the" here implies that there was only one girl who went camping with you; if there were in fact more than one, the clause "who went camping with us" would not be sufficient to identify the particular girl that you are referring to. If there were more than one girl, to justify using "the" you would have to add other specifying details, saying perhaps "Do you remember the girl from Iowa who went camping with us last May?"]

"John is reading a book about quantum physics." [Here the noun "book" is modified by the phrase "about quantum physics." Since there is undoubtedly more than one book about quantum physics, so "book" is indefinite and requires the article "a." In order to make "book" definite, we would have to add more information: "John is reading the book about quantum physics that was assigned by Professor Jackson last week."]

5) The context or situation makes the noun's identity clear.

For example, you might ask someone to "Close the door." You would use "the" because it would undoubtedly be clear to both of you which door you were referring to. (If it were not, if in fact there were several doors that could be closed, then it would be inappropriate to say "Close the door."
Similarly, if you tell someone that you are "going to the library," that person will assume that you are talking about whichever library is most familiar to both of you--RPI's Folsom Library, for example.

Again, you have to be sure that your reader or listener has the same context or situation in mind that you are thinking of; otherwise, he or she will be confused by your use of "the. "

For example, one student wrote the following sentence. "This magazine helps women analyze the problematic situation and offers possible remedies."

However, this was the first time she had mentioned a problematic situation. Her readers were therefore confused, because her use of the word the implied that they were already supposed to know which problematic situation she was referring to, and they did not.

Rather, the sentence should read, "This magazine helps women analyze a problematic situation and offers possible remedies" OR "This magazine helps women analyze problematic situations and offers possible remedies."