The Paradox is based on a fair game of tossing a coin, with the pot starting at one dollar and doubling each time a head appears. When the coin comes up tails you get paid and the game ends. So the odds and payoffs are are
50% chance of getting 1 dollar
25% of getting 2
12.5% of getting 4
6.25% of getting 8 etc.
Thus the expected value of the payoff is infinite, assuming that the casino has infinite money.
he paradox is how much you would be willing to pay for it, because rationally according to your expected payoff this should be infinite, but in reality people would not pay this, and research suggests they would only pay $20 or so.
It is so named after it was presented in St Petersburg by Daniel Bernoulli in 1738, although it was actually originally invented by his cousin Nicolas, found in a letter in 1713.
In terms of a recent example one can look at high growth stocks such as Facebook or Zynga. For early investors the potential pay off is huge (although not infinite). One difference is that once large growth rates start to tail off (for example going from 100% a year to 50%/yr) investors suddenly wake up to the paradox and sell stock.