Incorrect: We live in city.
Correct: We live in a city.

Explanation

A singular common noun (e.g. city, state, country, boy, girl, teacher etc.) takes the article a/an before it. But if the common noun refers to a particular person or thing it requires the definite article the whether the noun is singular or plural.

Compare: We live in a city. (Here we use the indefinite article because we are not referring to any particular city.)
The city is very big. (Here we use the definite article (the) because we are referring to a particular city that has already been mentioned in a previous sentence.)

Incorrect: He is best player.
Correct: He is the best player.
Incorrect: She is a most intelligent girl in the class.
Correct: She is the most intelligent girl in the class.

Explanation

Adjectives in the superlative degree takes the article the before them.

Incorrect: The London is big city.
Correct: London is a big city.
Incorrect: I live in the Mumbai.
Correct: I live in Mumbai.

Explanation

Both London and Mumbai are proper nouns because they are the names of particular cities. Proper nouns do not take articles before them.
Remember that a noun can be proper in one sentence and common in another sentence; so it is useless to label a particular noun as proper or common.

Incorrect: The gold is yellow.
Correct: Gold is yellow.

Explanation

Material nouns (gold, rice, silver, iron, wood, marble etc.) do not take articles before them.

Incorrect: Himalayas are mountains.
Correct: The Himalayas are mountains.
Incorrect: We should love the God.
Correct: We should love God.
Incorrect: The man is a member of society.
Correct: Man is a member of society.

Explanation

Here the noun man refers to the whole of mankind. We do not use articles before a noun used to refer to the whole of its kind.

Incorrect: We had a picnics nearly everyday.
Correct: We had picnics nearly everyday.

Articles are not normally used before plural common nouns that do not refer to a particular person or thing.

Incorrect: Each of us loves our country.
Correct: Each of us loves his/her country.
Incorrect: None of the boys had brought their books.
Correct: None of the boys had brought his books.

Explanation

The pronoun referring back to singular words like each, every and none should be singular in number. Note that this rule is no longer strictly followed. Sentences like ‘Each of us loves our country’ and ‘None of the boys had brought their books’ are now considered correct in informal speech and writing. However, in a formal style you must stick to the rules and use the correct pronoun.