Here is my attempt at some examples (IMPORTANT EDIT: My math below is wrong. See post #1 in this thread for correct math. I'm leaving my wrong math below for posterity):

1) The bitcoin ultra-bull: buys 500 puts at a strike price of .02 BTC/USD (The right to buy 10 bitcoins for $500) for 0.1 BTC. When bitcoins skyrocket to $1000 each, he exercises the put, in which case he gets to buy 10 bitcoins at $50 each when they are currently worth $1000 each.
2) The bitcoin ultra-bear: buys 5000 calls at a strike price of 0.5 BTC/USD (The right to sell 2500 bitcoins for $5000) for 1 BTC. . When bitcoins crash to 0.01 each, he exercises the call, in which case he gets to sell 2500 bitcoins at $2 each when they are currently worth 0.01 each.
3) The ultra-bull counter-party need not be a bear, but rather just someone who doesn't think that bitcoins are going above $50 before the option expires. They put their 10 bitcoins in escrow (currently worth < $200), and they either get 10.1 bitcoins back, or $500 USD + 0.1 BTC back. Not a bad deal.
4) The ultra-bear counter-party is just someone who doesn't think bitcoins are going to fall under $2 before the option expires. They put their $5000 in escrow (currently worth < 300 bitcoins), and they either get $5000 USD + 1 BTC back, or 2501 bitcoins

The counter-parties can make a nice steady income just betting against massive changes in bitcoin values. The ones buying puts and calls are essentially playing the lottery - betting on massive changes.

There are a LOT of other things that you can do with options (spreads and such), but this should get people started. Please help me fix any errors above - I am not an expert on options, and I may have gotten something completely wrong.