In America, both the President and Vice President are elected.  It's known when you vote that the VP will replace the President if the President cannot perform his or her duties.  So you go into the voting booth knowing that.  Apparently it's also a fine way of keeping a country from impeaching the President: pick a nasty VP...

And every 4 years there is an election.  

When Tony Blair was elected, did people know Gordon Brown would be his successor if Blair stepped down?  Or could Blair pick anyone he chose, so long as they were of the Labor party?

Also, I don't understand the term limits.  Do they have them?  Why could Blair step down and appoint a successor instead of having to serve out a term at the end of which would be a general election?  They have something like "confidence" votes, right?  To see if they can stay on?  And if they win, then the only thing that prevents them from stepping down and putting a friend in charge is a Parliamentary election?  Are those done on a regular schedule?  

"Pretending that you already know the answer when you don't is not actually very helpful." ~Migeru.