The Custap Berry is a currently unavailable Berry that allows the holder to 'move first just once in a pinch'. A pinch, as usual, is when the holder's HP is 1/4 of its maximum or less.
This has some obvious advantages and disadvantages when compared to Salac Berry. Salac Berry gives a one-level Speed boost, so it's usually used in conjunction with Substitute or Endure to set up a swep, possibly with Flail/Reversal. This strategy isn't as popular in D/P due to the popularity of Sandstorm and Steath Rock and the glut of new priority moves. Anyhow, the obvious disadvantage of the Custap Berry is that unlike Salac Berry, the boost is only good for one turn. The obvious advantage is that it works better on very slow Pokémon; ones that wouldn't be able to sweep even with a one-level Speed boost. However, this is not the only advantage of the Custap Berry.
Unlike most pinch Berries, the Custap Berry is not eaten immediately after the Pokémon is reduced to 1/4 HP. Instead, a check is made at the beginning of the round at the same time a Quick Claw check would be, and the Berry isn't eaten until right before the boosted attack is launched. What this means is that your opponent doesn't know the effect is coming until it's too late.
In order to take advantage of this element of surprise, it's best to abandon the traditional methods of activating a pinch berry. If you use Endure or Subsititute down to 1 HP and no Berry gets eaten, your opponent is going to realize that something is up. Instead, you need to pick Pokémon and situations in which your Pokémon will reach the appropriate thresholds without these aids.
Since the Berry isn't eaten right when the holder's HP drops, you don't even have to use the effect immediately. If you're afraid that your opponent has a priority move or some other way to block your attack, you could switch out your Pokémon with Custap Berry and use the boosted attack later, assuming it can still survive any entry hazards on the way back in. Alternatively, you can even use your opponent's entry hazards to reduce your Pokémon to the 1/4 HP threshold if you're careful about it.
Like Shadow Knight said, the primary uses of a Custap Berry are:
Destiny Bond: Slow, durable Pokémon with Destiny Bond that can reasonably expect to be reduced to the 1/4 HP range without the help of Substitute, Focus Sash, or Endure can take advantage of Custap Berry. Good candidates are Weezing, Wobbuffet, Swalot, Dusknoir, Vespiquen, and Spiritomb.
Explosion: Likewise, Pokémon that can take hits and use Explosion are good candidates for the Custap Berry. Examples are Golem, Muk, Cloyster, Steelix, Exeggutor, Lickilicky, Forretress, Camerupt, and Torkoal.
EDIT: Post changed now that the mystery of when Custap activates has been put to rest.