Robert Heinlein does a good job of describing a tesseract in his short story “And He Built a Crooked House”. You can read it here.

[slight hijack] I wonder, what would a 4th dimension sphere be like? Would it be called a hyper-sphere or some such? What about a hyper-pyramid or cone?

“I madam, I made radio! So I dared! Am I mad? Am I?” ~Nikola Tesla

Well, isn’t a sphere just an infinite number of 2D circles around a single point?

Yes, 4D sphere would be called a hyper-sphere. Just like the lower dimensional circle and sphere, it would be all the points a fixed distance from the center, but in a 4D space. Its hyper-surface would be 3-dimensional – unbounded yet finite 3-dimensional space.

It’s a little tough for me to imagine exactly what these would be like, but without a doubt they can exist. You might google for “hyper-cone” to see what you can find.

I think we can come up with reasonable definitions for hyper-cone and hyper-pyramid. A hyper-plane is a commonly defined as a 3D space imbedded in a 4D space. A (3D) sphere lies entirely in some hyper plane. Take a sphere in 4D space. From the center of the sphere, draw a line that is perpendicular to the hyper-plane that contains the sphere. Pick a point on this line; we’ll call this point the apex. A hyper-cone will be all of the lines connecting the apex to points on the sphere. A hyper-pyramid could be defined similarly using a cube as the base.

Not only is that definition reasonable, it’s actually quite commonly used…