Hyperspheres are spheres in a dimension N, greater than 3. In two dimensions we have a circle, in 3 dimensions we have our first sphere. It is very hard for us to visualize spheres in larger dimensions, since we leave in a 3D world.
Sphere packing is an interesting concept - filling N-D space with N-D spheres, which has led to exposed truths about 8 and 24 dimensions - truths which we don't even understand in 4 dimensions. We currently only know how to do sphere packing in dimensions 2, 3, 8 and 24.
Visualizing a hypersphere is very hard. An N-th dimensional space simply has N coordinates. For example a 4D dimensional space has 4 coordinates: (x, y, z, w). One way to visualize a 4D sphere, for example, is to use slicing. For example, if we move a 3D sphere through a 2D plane, we will obtain a 2D slice of our original sphere. Similarly, if we take a 4D sphere and pass it through a 3D cube, we will obtain 3D slices of our 4D sphere, which can be visualized.
Hyperspheres a very counterintuitive. For example, consider a circle inscribed in a square, so that it touches the squares walls. It covers 79% of the square's surface. If we consider the case in 3 dimensions, a sphere only occupies 52% of the area inside of a cube. If we keep going, in 30 dimensions the sphere will occupy only 10 to the -13 the size of the cube inside it. This is about as big as a grain of sand is to a sports arena, only that somehow the sand grain is touching each wall of the sports arena and is still round.
For more information and some nice visualizations please see the video below, by PBS Infinite Series:
