I didn't get the memo

Can somebody explain to me how come Tony Blair resigns and some dude called Gordon Brown, Chancellor of the Exchequer* (love this word, exchequer) just scoots on over and replaces him? Aren't people supposed to vote? Is the Prime Minister's office an hereditary mandate? Ignorant Enchiladan wants to know.
I know this may not endear me to some of my British friends on the other side of the pond, but I like Tony Blair. He could say several sentences in beautiful English, which is much more that we can say for what we have here. He is capable and intelligent and a leader, something we pine for over here, some sort of a miracle around these parts lately. Also, he was great in The Queen.
It is very sad that he will be remembered with bitterness for being loyal to an idiot who destroyed pretty much everything else he had accomplished. Certainly, he is solely responsible for not standing up to Bush and reading him the riot act. He could and should have. What was he thinking?

*Cf. "Chancellor of the Exchequer" with "Secretary of the Treasury". First one sounds so much better, sexier, snappier, less pedestrian, although I always imagine some guy in a powdered wig signing away checks in the Tower of London.