Barrier Options

A barrier option is  a normal option with an embedded feature that causes it to be activated or terminated when the barrier is breached. It is attractive to investors because it offers a cheaper hedge compared to a vanilla option due to existence of the barrier. However, barrier options can be bad hedges especially in times of high volatility in market.
As the payoff of a barrier option changes abruptly as the underlying crosses the barrier, the option experiences high delta sensitivity (high gamma) in the vicinity around the barrier. Also, unlike a normal option whereby the buyer is long vega, buyer of a knockout option can be short vega near the barrier. This is because when spot moves closer to the barrier, a higher volatility can increase the chance of hitting the barrier and thus the option becoming worthless.
A simple illustration: Suppose you buy a knockout USD call/JPYput strike at 85 with a barrier of 75 with expiry 3mth. The option extinguishes if USD/JPY ever trades below 75 during the life of the option. The reason  you would want to buy this is because you believe that if USD/JPY ever trades below 75, it is unlikely to rise above 85 at expiry. And you pay a lower premium for this as a result of the barrier. However, in the event that USD/JPY falls below 75 prior to expiry but above 85 at expiry, the option would already have terminated and you would not get the payoff.
A reverse knockout terminates an in-the-mony option. In the above example, the barrier now moves to 95 instead of 75. So if spot is at 94, you will still get a profit. However as spot crosses the barrier, your deep-in-the money option becomes worthless. It's common to add rebate to a knockout structure like this as a compensation (one-touch digital).
Knockouts in FX have been very popular prior to the 2008 subprime crisis. It allows investors to buy protection cheaply. However, the high volatility that followed had caused many barrier options to be terminated abruptly. And this has left investors with no protection as per the initial investment.

Digitals..
The simplest form of barrier. A digital call is when spot moves above strike, the buyer gets a fixed payoff regardless of the location of the spot from the barrier; otherwise payoff is zero. A digital has high delta sensitivity in the vicinity of the barrier, so delta hedging is difficult. The call digital can be replicated (approximately) by a call spread (buying a lower strike call + selling the higher strike call; with both strikes very close to each other).

Asset-or-nothing..
Asset-or-nothing call pays the value of the underlying if spot moves above a certain level. This can be replicated by a vanilla call + digital at the same strike.