Singular count nouns cannot stand alone in a sentence.

Apple is on table. (Wrong!)
They must be preceded by one of the following:
1. An article (the, a, an)

An apple is on the table.
The professor gave us a test.
2. A number or quantifier (one, another, the other, each, every, either, neither)

One man was in Chicago.
Neither man was with his wife. The other man was in Saint Paul.
Either man could be in trouble.

Note: Never use the and another, each, every, neither, either together:

The another man was in Milwaukee. (Wrong!)
3. This, that, some*

This artist created that painting.
Some woman came to see you. (*unidentified subject)
4. A possessive p___oun or noun.
It is my bicycle now.
It was John's bicycle.
Note: Do not use articles and possessives together.

The Mary's bicycle was stolen. (Wrong!)
Proper nouns have their own article usage.
Don Smith lived next door to Dr. Zimbango.
The Empire State Building is in New York City.
Non-count and plural nouns can be used without articles only in the ?generic? sense:
Cats are enemies of dogs.
Water is essential for survival.

Article Chart

Specific
(This one, that one)
(This/that group)
Which one?
Which ones? Non-Specific
Any one
Any group
One of many
One of many groups Generic
In general

Count
Singular The apple
The bird
The child An apple
A bird
A child *
*
Count
Plural The apples
The birds
The children Some apples
Some birds
Some children Apples
Birds
Children
Non-count
The water
The information Some water
Some information Water
Information

Notes:
Specific articles are used with nouns which have been identified previously. (The speaker and the listener both know which thing/person/substance/idea is being referred to.)

The teacher is coming up the stairs.
(Both listener and speaker know which teacher and which stairs.)

Give me the red shirt. (I know which one you are talking about.)

Non-specific articles are used with nouns that have not been identified previously (by both the speaker and the listener.) They are used with items that have not been singled-out yet. (Note: As soon as the items are identified, they require a specific article.)
I want a candy bar. (Any candy bar will do.)
Which one do you want? (Asking for specification)

The one on the right. (I choose that one.)

Give me some milk. (Any milk is fine.)

I need some new shoes. (But I haven't decided which ones to buy yet.)

I bought some shoes at Valmart. (I know which shoes, but you don't.)

These are the shoes that I bought. (Now we both know which ones.)

Non-count and plural nouns are used without articles in the generic sense.
Cats are afraid of dogs. (in general)
Water is necessary for survival.

*However, singular count nouns cannot stand alone in a sentence, so an article (usually a or an) is used.
Oranges contain Vitamin C. (generally)
Orange contains Vitamin C. (incorrect)

An orange contains Vitamin C. (okay)

See also:
JustSomeLyrics
77
77.109
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Don McLain American Pie Lyrics