If you can reasonably count the individuals of the group it is 'number', but if you cannot it is 'amount'. This is the rule I taught my children.

I think the the comment made above regarding countable and measurable as a basic method of distinction between the two is a good model to follow.

I do dislike hearing a group of people refered to as an 'amount of people'. Not nice! I think I need one big Bloody Mary and that would be a large amount of Bloody Mary. Then again I could recover more slowly by consuming several individual Bloody Mary's but that number of Bloody Mary's would render me even more senseless than I already am. :-)