It gives a bit of grind to your sound, yes, but not outright overdrive.
It'll work well if your bass normally has a loud output, not so if it's not a high-output bass.
Depending on how you wire it, you could have it on at all times, together with a regular tone control or instead of a regular tone control. If you put it in place of a tone control and you dial it up to maximum, it won't have any effect on the sound.
And here's the best bit. The Black Ice Overdrive is two diodes epoxied and sold at what I'd say is a ridiculous price, and you can easily make one yourself.
Go into a Radio Shack or similar, pick up two Schottky diodes and two standard diodes. If you've never worked with them, diodes are small components that look like a miniature plastic cylinder with a stripe on one end, and under normal circumstances only allow the current to go in one direction, while they block the other way round.
To install it, desolder your tone capacitor, solder one on in the forward direction (clean end to the tone pot's lug, striped end to the pot casing or wherever the ground is), then solder one in the reverse direction (striped end to the same lug on the tone pot, clean end to ground). Now they will be connected in anti-parallel, which is how they should be if you want it to act like a regular Black Ice. Soldering's done and you can feel free to try it out.
If the instructions were a bit unclear, check this image out
The reason why I said you should get two sets of diodes is because all diodes have a certain voltage at which they start clipping. Schottky diodes react much sooner (i.e. to much lower voltages) than regular diodes. Try both sets, see which you like more, and if you like the effect at all.
If you want to, you can also try leaving your tone cap on and in parallel with the diodes, or connect the tone cap inbetween the joined ends of the diodes towards the tone pot. Or even run the diodes in anti-series (one after the other, facing opposite directions, although this works better with Schottkies). Or even mix-and-match the diode types in any of the above combinations.