What surface can we slice to obtain a cubic curve? We all know what a conic section is - a circle, a hyperbola, an ellipse, or a parabola.
But what about the cubic curve? Does it not slice through some other 3D shape? If so, what is that called? What other curves can we find in it?
If there IS no answer, or mathematicians of this day and age haven't found it yet, please let me know in these comments, as I'm still very very curious. I'd like to know if it doesn't exist!
 A: Any (algebraic) curve can be thought of as a slice of a 3D shape. Suppose your curve is defined by an equation $f(x,y)=0$: then define a polynomial $F(x,y,z)$ by some formula
$$F(x,y,z) = f(x,y) + z \cdot ( \text{ anything you like} ). $$
Then the equation $F(x,y,z)=0$ defines a 3D shape whose slice $\{z=0\}$ is exactly the curve you started with.
In the particular case you ask about, the nicest answer is to view cubic curves as slices of cubic surfaces. If you search for that term, you'll see that cubic surfaces are much-loved by algebraic geometers for their beautiful, regular geometric properties. (Warning: to get the nice answer, one needs to think of surfaces in projective space.)
For the last part of the question, there are lots of other kinds of curves inside cubic surfaces. For example, probably the most famous property of these surfaces is that they always (again, in the projective world) contain exactly 27 lines. If we take a slice by a plane containing a line, we get the line together with a conic. Any slice by a plane that doesn't have one of the 27 lines in it will give a plane cubic curve, but if you look at the picture in the affine plane over the real numbers (as in the original context of your question) there is a multitude of types of these. (Newton classified them and found something like 72 distinct types, but if memory serves he missed a few cases.) One could think of this big list as the analogue of the four types of conic sections you mention. 
Finally let me mention that both in the conic and cubic case, you can get many, many other curves in your surfaces by taking slices by things other than planes --- that is, intersections with surfaces of higher degree. But that is probably taking us too far from the original question. 
