There are two main articles, the and a/an. How articles are used depends on whether the noun is a singular, plural, proper or a mass noun, and whether the noun has been previously mentioned.
“A” and “an” introduce a noun that has a quantity of one. In this case, “a” means the same thing as “one.” If the quantity of an object is greater than one, neither “a” nor “an” is used.
“There is A cat stuck in the apple tree.”
“There is ONE cat stuck in the apple tree.”
“There are TWO cats stuck in the apple tree.”
Neither “the” nor “a” is used when referring to “proper nouns,” such as names of people, places or things (one exception is “The Netherlands”). Articles are also not used with “generic nouns” or with “mass nouns.” Mass nouns are usually things composed of smaller things, such as some kind of food (a single piece of rice, for example, is “a grain of rice,” not “a rice”):
“Boston is my favorite American city.” (“Boston” is a proper noun.)
“Firemen are very brave.” (“Firemen” is generic because it refers to all firemen, not any specific one.)
“Farmers grow wheat” (“Wheat, corn, rice,” etc. are mass nouns.)
PREVIOUS MENTION can also determine what kind of article a noun will take. The use of “a” or “an” implies that the noun is being mentioned for the first time. Once the noun has been mentioned, however, it is referred to with “the.” Note how “a” and “the” are used in the following sentences:
There was a cat [first mention] stuck in a tree [first mention]. The cat [second mention] was rescued by a policeman [first mention]. The policeman [second mention] fell out of the tree [second mention] and broke his leg.
Some nouns take “the” instead of “a,” even when mentioned for the first time, because the noun is assumed to be commonly understood.
“The president of IBM addressed the convention.” (The president is identified as the president of IBM.)
“The president addressed Congress.” (Here, the president is the president of the United States.)