Number is used with plural nouns, things that can be counted.
I saw a number of pigeons on the shed this morning.
Amount is normally used for nouns that can't be measured.
The amount of animosity generated by his comments was out of all proportion to his words.
But it can also be used for things that can be measured (as @psmears points out below), especially money:
Please send your payment in the amount of $253.79.
Quantity is used for nouns you can measure.
I have a large quantity of pumpkins for sale.*
It is pretty much synonymous with number.