Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
 noun Any of a set of objects resulting from the generalization of a twodimensional circle and a threedimensional sphere to n dimensions. In ndimensional space, a hypersphere is the set of all points that are a given distance, called the radius, from a given point, called the center.
from The Century Dictionary.
 noun Equidistantial surface.
 noun In fourdimensional space, the threedimensional quadric spaces through the intersection of X_{5} = 0 and where is a threedimensional space which is met by any line of S_{4} in two points.
from the GNU version of the Collaborative International Dictionary of English.
 noun A mathematical object existing in more than three dimensions, analogous to the sphere in that all points on the surface are equidistant from the central point; a generalization of a sphere in more than three dimensions.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun geometry The set of all points in a given
hyperspace that are at a given distance from a given point.  noun sports A
zorb ; the act ofzorbing , entering a zorbing ball, strapping into a harness, and rolling down a hill.
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
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Examples

Sometimes the term hypersphere is used to denote the hypersolid, the portion of fourspace inclosed by this locus, which is then called the boundary or hypersurface of the hypersphere.

Either that or it has something to do with the hypersphere as described by Poincaré, making the universe finite but unbounded.

Its probably more like a finite hypersphere 4dimensional spherelikething.
Astronomers Find Black Holes Do Not Absorb Dark Matter  Universe Today 2010

It is relevant to discussions between Sam, myself, Lawrence, John, and a few others earlier in the thread, considering a hypersphere universe and the merging of GR/SR/QM.
What’s the (Dark) Matter? John 2008

Make arbitrary Earthstyle coordinates for a hypersphere.
A Dark, Misleading Force Sean 2007

~ Also from Joe: Kosmic Coordinates on the Kronos mandala or, the birth of a hypersphere.
Speedlinking 2/28/07 William Harryman 2007

If the mansion stood on a hypersphere slightly greater in diameter than the mansion grounds, a person could move from any point to any other with what, in threespace, would seem to be right angles.
Orphans of Chaos 2005

For a moment, I gave off a shaking shock wave similar to what the hypersphere had done; but something was carried with it.
Orphans of Chaos 2005

The shock waves the hypersphere gave off were only "sound" along one surface (hypersurface?) of the concentrically expanding ripples of pressure.
Orphans of Chaos 2005

She never heard the ringing in a locked safe of the hypersphere shaking from the shock of the music of a siren.
Orphans of Chaos 2005
bilby commented on the word hypersphere
Extremely spherical.
July 19, 2009
sionnach commented on the word hypersphere
in contrast to the hyposphere, which is manifestly deficient in sphericity, a bit like a hedgehog being stretched on the rack by some tziganoporcaphobic Torquemada.
July 19, 2009
john commented on the word hypersphere
Reminds me of Rounder Records.
July 19, 2009