If you slice a torus (a doughnut-shaped surface) in half with a plane parallel to its axis, the cross section is two circles.
Likewise, if you cut the torus across the middle with a plane perpendicular to its axis as if you were slicing a bagel, you would see two circles: one circle going around the circumference and a smaller circle defining the hole. There is one more pair of circles hidden inside a torus that goes through any point on the torus. If you cut the torus in half at just the right slant, your cross section will be two interlocking circles called Villarceau circles. The animated image below (from Wikipedia) shows how this is possible.
The paper model in this project shows Villarceau circles in a torus. Each seam in the model is one of the two Villarceau circles that goes through a point. The name Clifford torus refers to the fact that this torus is a stereographic projection of a 4D Clifford torus into 3D.