What kind of power do the British royals have, and where do they get their wealth?

I'm a clueless American who is baffled by the fact that there are modern societies that still have some sort or rulers based on blood line. I'm assuming political power belongs entirely to the Prime Minister and his crew (when and how did that invade the monarchy?) and that taxes are collected for that government. What's the point of the royals today (other than tabloid news) and where do they get their wealth?