The Monty Hall problem is a pretty cool one. I think the easiest way to understand it is by realising:
1.) After choosing Door A, the probability that the car is behind Door B or Door C combined is 2/3.
2.) After the host reveals that Door B or C has a goat behind it, the 2/3 probability which was previously distributed over 2 doors is now shifted to one door (B or C, whichever one the host didn't open). So if we switch to this door, there's a 2/3 prob that the car will be behind it, but only a 1/3 prob that the car is behind Door A which we originally chose.
My favourite is Russell's Paradox which goes like this:
We define set S as the set of all sets which don't contain themselves as members. The paradox arises when you consider whether S should contain itself. If S contains itself as a member, that's a direct contradiction of its definition. And if S doesn't contain itself, then it should by its definition as the set of all sets which don't contain themselves.
The common non-mathematical analogy is the Barber's Paradox where we have a barber who shaves
everyone who does not shave themselves... this is right-track's ideal town where no-one has a beard
The paradox of course presents itself when we consider whether the barber should shave himself.