http://unsuck-it.com/as-soon-as-possible-asap/
Strange; Whenever someone tells me they need something ASAP, I generally consider it to mean "they don't know when they need it, so it's not important".
You COULD make an artificial deadline and say "I need this today." But that means "drop everything for me." Maybe you're emailing your boss and don't have the clout to demand that, or maybe you want to save that demand for when you REALLY need it today.
So ASAP would be my way of saying "it's important, but I recognize that something else on your list might be even more important." The "P" gives the receiver some wiggle room. In that way, I think it's more polite than "today," while still more urgent than "sometime this week."