To construct the points of the intersection of two cones we choose a system of
cutting planes that intersect both cones along their generatrices.
A plane through the vertex of a cone intersects that cone in two generatrices. Therefore, the system of planes that intersect both cones in their generatrices is a pencil of planes that contain the line through both vertices of the cones. Every plane of that pencil intersects each of the two cones in two generatrices. These generatrices intersect in 6 points. Four points which are intersections of these four lines, different from the vertices of the surfaces, are points on the curve of intersection. 
The construction of the intersection curve is detemination of all real points of that curve.
The cutting plane can intersect a cone in two real and different generatrices, in one generatrix when the plane is a tangent plane and in two imaginary generatices. If a cutting plane intersects both cones in one real generatrix, this plane is a common tangent plane and the intersection of these two generatrices is a double point of the intersection curve (as is shown in the figure). 
If one cone passes only partially through the other cone and on both cones exist generatrices that do not intersect the other cone in real points, then the intersection curve is onebranch. This kind of intersection curve is called partial intersection.  If one cone passes completely through the other cone and all generatrices of that cone intersect the other cone in two real points, then the intersection curve is twobranch. This kind of intersection curve is called complete intersection. 
PARTIAL INTERSECTION  a onebranch curve 
COMPLETE INTERSECTION  a twobranch curve 
Two cases when the intersection curve has a double point:
When two cones have a common tangent plane in a common regular point, that point is a double point of the intersection curve.  If one cone passes through a double point of the other cone, its vertex, that point is a double point of the intersection curve. 
an intersection curve with a DOUBLE POINT In this double point the cones have the same tangent plane. 
an intersection curve with a DOUBLE POINT Double point is a vertex of one of the cones. 
The case of total degeneration into 4 lines appears when two cones have the same vertex. In that case, the two cones have 4 common generatrices that pass through the common vertex and through the four intersections of the base curves of the cones. These four intersections can be real and different, real and coincide or pairs of conjugate imaginary points and their type determines the type of generatrices. Every two cones with a common vertex intersect at four lines that pass through the vertex. 
The intersection curve of two cones can never degenerate into two different lines and a conic because in that case the intersection point of the two lines would be the vertex of
the cones. As we have seen, if the cones have a common vertex, their intersection curve degenerates into 4 lines. However, if the two cones have a common generatrix and a common tangent plane along that generatrix, then this generatrix is a line counted twice that appears in the degeneration of the intersection curve, together with a conic. When two cones have different vertices but one common generatrix with a common tangent plane along that generatrix, then the intersection curve degenerates into a double line and a conic. 
four lines 
a double line and a conic 
If two cones have one common generatrix but different vertices and different tangent planes along that common generatrix, then their intersection curve degenerates into a line and a curve of the 3rd order.

If two cones have two common tangent planes but different vertices and do not have a common generatrix, then their intersection curve degenerates into two conics. If the cones are cones of revolution, this kind of degeneration of the intersection curve appears only if their axes are intersecting lines. 
a line and a curve of the 3rd order 
two conics 
Created by Sonja Gorjanc, translated by Helena Halas and Iva Kodrnja  3DGeomTeh  Developing project of the University of Zagreb