The frequent confusion between amount and number is based on a misunderstanding of a small, simple distinction that the words themselves indicate.
Amount refers to quantities that are measured in bulk or mass — considered as a whole — while number pertains to things that can be counted individually: “The amount of square footage in the attached shed is enough for a push lawn mower or a small barbecue,” but “The total number of square feet includes the attached shed’s square footage.”
In reference to amount, use the terms little or less or the phrase “much more”:
“The amount left is too little to be of any use.”
“The amount is less than I thought.”
“The amount is much more than we expected.”
In reference to number, use the words few or fewer or the phrase “many more”:
“The number of people who have signed up is too few.”
“The number of people here, compared to the number here yesterday, is fewer.”
“The number of people here is many more than we expected.”
(Note that more applies to both amount and number.)
Either word can be applied to a particular thing as long as the description of the thing is consistent with the distinction between amount and number: One can refer to the amount of fun one has had, but one can also count the number of fun things one has done.
Two categories of things that are flexible in terms of these usages are money and time: One can refer to an amount of money or to a number of dollar bills, or to an amount of time or a number of hours: “The amount of money in the cookie jar has decreased” and “The number of dollar bills in the cookie jar has decreased” are both correct, as are “I need to decrease the amount of hours I’m scheduled to work” and “I need to decrease the number of hours I’m scheduled to work.”