Down-and-In Option
  
A down-and-in option is a kind of knock-in option, which is itself a kind of barrier option.
Don't know these terms? Let's start with the barrier part.
A barrier option only exists if the underlying asset reaches a certain price (or barrier). Shares of Chicago Urban Windmills Inc. are trading at $25. You buy a barrier option that only kicks in if shares reach $30. At the end of the option expiration, the stock has only reached $29. It's as if the option never existed. It didn't reach the barrier, so the option never kicks in.
Okay, the knock-in part. That means that the option only comes into existence if the underlying asset reaches a certain price. The Chicago Urban Windmills example works here too. It needed to reach $30. It only reached $29, so the "knock-in" didn't take place.
The knock-out option represents the opposite of the knock-in version. (A knock-up option is a completely different thing...one that would probably involve a discussion with an OB-GYN.)
In the knock-out case, the option is live, unless the price of the underlying asset reaches a certain level. Then it gets turned off, instead of getting turned on. So a knock-out option involving Chicago Urban Windmills would go like this:
Shares are trading at $25. You put a knock-out option into play with a barrier at $30. The option is still in place at $29. But if shares rise to $31, the knock-out option is no longer valid. You got...knocked out.
That explanation gives you the "in" part of the down-and-in option. It's a knock-in option, meaning it only turns on if the price of the underlying asset moves past a pre-set barrier.
The "down" part? Pretty much what you think. The down-and-in option involves the price of the asset going...down.
So, Chicago Urban Windmills is trading at $29 now. You think it will go back down. You set a down-and-in option with a barrier at $27. If it falls to $28, that does you no good. Your option doesn't apply. However, if it falls to $25, you're set. It has fallen past the barrier, thus activating the option.