The same principle applies to amount and number as applies to less and fewer.

Amount (like less) is used for things that cannot be counted:  an amount of rain, a considerable amount of courage, a large amount of trouble.  We should not be confused by "an amount of money."  True, we can count money, but we don't count "one money, two moneys," etc.

Number (like fewer) is used only for countable things:  a number of raindrops, a number of brave men, a number of problems, a number of pennies.

The most common error with respect to amount and number is with the erroneous use of amount to refer to people.  "A large amount of people" or "a large amount of students" is incorrect.  We should use "a large number of people" or "a large number of students" because we can count people (one person, two people, etc.) and students.