1. We use amount with things we can't count (or do not usually count).
Ex - an amount of water, an amount of rice, etc.

2. Amount is also used with abstract concepts.
Ex - The amount of love people feel for each other, The amount of pride etc

3. The word number refers to a quantity of something that can be counted.
Ex - things we can count are people, apples, mobile phones, photographs.

4. Sometimes we use the expression "a number of" when they want to avoid being specific about the quantity in question. In this case, the expression means "some" or "several.
Ex - Madonna's fans number in the millions.