Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A geometric figure in four or more dimensions, which is analogous to a cube in three dimensions. Specifically, the ndimensional equivalent of a cube for any nonnegative integer n.
 n. Such a figure in four dimensions; a tesseract.
 n. A computer architecture in which each processor is connected to n others based on analogy to a hypercube of n dimensions.
from the GNU version of the Collaborative International Dictionary of English
 n. A mathematical object existing in more than three dimensions, analogous to the cube in that each twodimensional facet of the surface is a square; a generalization of a cube in more than three dimensions.
Etymologies
Examples

Abstract: A representation of the genetic code as a six – dimensional Boolean hypercube is proposed.

Hyperspace means a space with more than three dimensions, and a hypercube is a cube in more than three dimensions.

The hypercube is a very particular case of the hyperprism.

For example, if we want to visualize the 4dimensional equivalent of a cube (called a "hypercube", with 16 nodes and 32 edges), we can get a feel for it by "stereographically projecting" it into 3dimensions:

Twitter search for "hypercube" and you'll see the naysayers like

: P matociquala: It might be a hypercube and have LOTS of corners hawkwing_lb: (the one shape heaven shouldn't be is a pyramid)

Silently, he ushered him through a door marked with the twodimensional shadow of a hypercube.
365 tomorrows » Sam Clough : A New Free Flash Fiction SciFi Story Every Day

Gwiazda has created a kind of 4th dimensional hypercube format for some of these images, which allows him to imagine that he can simultaneously see in front, to both sides, and behind him.

Interesting result from graph theory (and the theory of parallel computing) is that any balanced colorcoded tree (see figures 4&5?) may be embedded in a hypercube of the same dimension w/o “fudging.”

The structure of the code facilitates evolution: the variation found at the variable positions of proteins do not corresponds to random jumps at the codon level, but to well defined regions of the hypercube.
asativum commented on the word hypercube
In mathematics, I believe a hypercube is a fourdimensional surface; in one sense it is to an ordinary cube what a cube is to a twodimensionsal square. Like other 4D objects, when rotated in 3space, it appears to change shape, but I'm pretty sure that's the only way time is involved. You can perhaps imagine your way to the timealtering speculation from here.
May 22, 2008
whichbe commented on the word hypercube
Based on the Tesseract model. The concept of a tesseract, or four dimensional cube, is often used as an illustration of time travel, and might also give a clue to how a warp in time might occur. As the tesseract constantly moves, collapses in on itself and then expands again in a never ending loop, so does time move forward in a never ending line. If, however, an object or person was able to move from one line of this tesseract to another at any junction point, they would in essence be moving not with time, but in a straight line perpendicular to it. This movement has led many science fiction authors to juxtapose the thought not of time travel, but of being able to stop time and then start it again.
(Wikipedia)
May 21, 2008