Take the cube centered at the origin whose vertices are $\lbrace (1 ,1 , 1) , (-1 ,1 , 1) , (1 ,-1 , 1) , (1 ,1 , -1) , (1 ,-1 , -1) , (-1 ,1 , -1) , (-1 ,-1 , 1) , (-1 ,-1 , -1) \rbrace$.
We can triangulate this cube using 48 simplices each of volume $\frac{1}{6}$. These simplices are congruent since we can obtain one from the other from a series of reflections, rotations, and translations.
What do we call these simplices? And their higher dimensional analogues?
EDIT: I'm looking for the terminology used to name these simplices in such a triangulation of an $n$-dimensional cube.
You seem to be describing a Barycentric Subdivision, in particular of a cube, which is even mentioned briefly on the Wikipedia page.
As is mentioned, each of the simplices is uniquely determined by a flag (sequence of nested faces, for example $\rm vertex \subset edge \subset 2$-$\rm face$) of the cube, and there are $48$ of these. In this case, since the cube is a regular polytope, its symmetry group acts transitively on the set of flags (hence simplices in the Barycentric Subdivision). As this symmetry group is a reflection group, I would probably call each individual simplex a chamber, although usually this is reserved for the (pieces of) subdivisions of each facet (in the case of a cube, each square face is divided into $8$ triangular chambers). As these are in bijection with the simplices, I think "chamber" is a good word.