Spherical panoramas are called spherical because that is the shape that the lens traces out as it rotates to capture the scene. Since you're essentially capturing a sphere of an image, then you have to "project" it onto a flat surface to display it as a flat image. This is exactly the same process that is used to create flat maps of the Earth. In fact the most well-known map projection, the Mercator projection, can be used for spherical panoramas.
Equirectangular or "Normal"
The projection that seems to be the most common for presenting a "normal" spherical panorama is the equirectangular projection.
Stereographic / "Planets" & "Tunnels"
The projection that I tend to use most often is the stereographic projection. There are several ways to use this.
By centering the image on the spot directly below the camera position you can create a "little planet" type of an image.
By centering on the spot directly above the camera position you can create a "tunnel" type of an image.