The whole zoo — ellipsoids, saddles, cones and both hyperboloids — and the traces that tell them apart.
Every one of these is the same equation with different signs. Change a sign and you change the animal.
Change one coefficient's sign and watch the shape become another.
A quadric is just the graph of a second-degree equation in three variables, and there are only a handful of them. What separates one from another is almost entirely the pattern of signs: all three positive gives a closed ellipsoid, one negative opens it into a hyperboloid of one sheet, two negative breaks it into two pieces.
The way to recognise one without plotting it is to take slices. Set z to a constant and see what curve is left; do it again for x and for y. Ellipses in every direction means an ellipsoid. Ellipses one way and hyperbolas another means a hyperboloid. Ellipses one way and parabolas another means a paraboloid.
The saddle is the one worth lingering on. Slice it vertically one way and you get a parabola opening up; slice it the perpendicular way and you get a parabola opening down. That single point at the middle is a maximum along one direction and a minimum along another — which is exactly what a saddle point of a function is, and why the second derivative test needs a determinant rather than just a sign.
The coefficients a, b, c only stretch the shape along each axis. They never change which animal you have; only the signs do.