Is there any criteria for expressing equations as graphs and vice-versa?

Well, all I know is that equations can be expressed as graphs and graphs can be expressed as equations. This is something we all learn at school but can anyone tell me why there is such a relation? And what is a criteria for such a relation to exist? (Also from @treble's comment) Why there is a need for such a criteria ("the vertical line test") to exist in the first place?

Also, is there any way to visualize a graph given a equation or just do the opposite (i.e given a graph, form an equation). It might be obvious for some equations like $y = x$, which is basically a line with a slope of $45^\circ$. However, what about equations like $y^2 = x^3 + ax + b$ (which is an equation of a Elliptic Curve) for which the visualization is not so obvious.

My question might be similar to this one but I guess I am asking more (or also, possibly, stupid-ish). Also, when meaning graphs and equations, I mean a 2D or 3D graph and polynomial equations.

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You should be more careful with your first sentence. The equation $x^2 +y^2 = 1$ cannot be expressed as a graph of a function, for instance, since it doesn't pass the "vertical line test." There is something called the implicit function theorem which gives you a criteria for when an equation can locally be expressed as a graph. –  treble Oct 13 '13 at 20:19
Thanks about the valuable insight. I rephrased the question. I hope that makes sense. –  TheRookierLearner Oct 13 '13 at 20:29