Continuous double concave is exactly what it sounds like: a double concave that runs uninterrupted from the nose all the way to the tail, without transitioning to single concave up front or to vee in the back. The most well-known example is the Slater Designs / Tomo I-Bolic platform that Kelly Slater uses across the world tour.
What it does on a surfboard
Continuous double concave maintains drive and speed through every part of the ride. There's no contour transition under your front foot or back foot — the board is "always on" in terms of channeling water past the fins. Slater's quote about it is that it's "always on," which captures the feel: maximum sustained speed, minimum need to pump or generate.
The trade-off is that the board can feel less reactive in tight pockets — there's no contour break to release water cleanly off the tail, so pivot turns require more deliberate weight shift. It's a contour built for drawn-out lines and down-the-line speed, not radical pivot snaps.
What to look for
Lundquist's I-Bolic-leaning models (notably the Wanted) run continuous double concave. Models built for pivot release through the tail (Innuendo, Five Horizons) run the more conventional single-to-double family. Pick the contour based on how you like to surf, not on what's fashionable.