Each normal disk is either a triangle which cuts off a vertex of the tetrahedron, or a quadrilateral which separates pairs of vertices.
This is either an octagon that separates pairs of vertices, or an annulus that connects two triangles and/or quadrilaterals by a tube.
The concept of normal surfaces is due to Hellmuth Kneser, who utilized it in his proof of the prime decomposition theorem for 3-manifolds.
The notion of spun normal surface is due to Bill Thurston.
Regina is software that enumerates normal and almost-normal surfaces in triangulated 3-manifolds, implementing Rubinstein's 3-sphere recognition algorithm, among other functionalities.