How did people figure out that parabolas, hyperbolas, circles, and ellipses were conic sections? Maybe it is not surprising if one thinks that parabolas, hyperbolas, circles, and ellipses are relatives because they all have kind of the same form of equations, i.e.,
$$
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,
$$
where different values of $A, B, C, D, E,$ and $F$, as well as their relations, give different kinds of curves.
But how did people figure out they were all conic sections (how could one see a cone from that second-order equation)?
Or was it the case that someone were playing with cones and somehow wondered how a cone could be cut in different ways and figured out that the edges from those cuts were all related through the above equation?

I think they knew it long before analytic geometry. 
You can prove those things using elementary geometry. The proves are elegant, but they  rarely appear in modern books; however, you can still find them in some comprehensive textbooks or online. For example, http://www.cut-the-knot.org/proofs/conics.shtml
But I'm not aware of the exact proves or reasoning that ancient Greeks used.  
