There are actually 3 different things here: graphs of functions, "Cartesian Equations", and parametric equations.

The distinction between graphs of functions and "Cartesian Equations" (to use your terminology) is best seen by considering a circle.

The equation \(\displaystyle x^2 + y^2 = 1\) is satisfied by a set of points (x, y) in the plane that form the unit circle at the origin. But it's not the graph of any function y = f(x).

(Would f(0) be 1, or -1? It can't be both for a function f. This is the origin of the so-called "vertical line rule", which the circle fails.)

So by "Cartesian Equations", I assuming you're including things like \(\displaystyle x^2 + y^2 = 1\), which is not the graph of any function. (It's the union of the graphs of 2 different functions, \(\displaystyle g_1(x) = \sqrt{1 - x^2}\) and \(\displaystyle g_2(x) = - \sqrt{1 - x^2}\).)

The following are the 3 types of sets in the plane it seems you want to consider:

1. Graphs of Functions: \(\displaystyle \{ (x, f(x)) \in \mathbb{R}^2 \ | \ x \in D_f \}\) (where \(\displaystyle D_f \subset \mathbb{R}\) is the function's domain.)

2. "Cartesian Equations": \(\displaystyle \{ (x, y) \in \mathbb{R}^2 \ | \ F(x, y) = 0 \}\)

3. Parametric Equations: \(\displaystyle \{ Q(t) = (x(t), y(t)) \in \mathbb{R}^2 \ | \ t \in D \}\), where \(\displaystyle Q : D (\subset \mathbb{R}) \rightarrow \mathbb{R}^2\).

The differences? Well, every graph of a function y = f(x) could be represented by a "Cartesian Equation" or a parametric equation.

Ex: \(\displaystyle F(x, y) = y - f(x)\) when x is in f's domain, and define \(\displaystyle F(x, y)\) however you like, other than zero, otherwise. And for parametric, set \(\displaystyle x(t) = t, y(t) = f(x(t))\), where the t-domain is f's domain.

However, not every "Cartesian Equation" can be

*continuously* parameterized.

\(\displaystyle F(x, y) = xy - x^3 = x(y - x^2)\), whose graph is the union of a parabola and the x-axis.

For practical purposes, the same is true in reverse (the formal claim would be a mess, and likely be false), meaning that not every parametric equation has the same image as a "Cartesian Equation"

Ex: Try to find F such that \(\displaystyle F(x(t), y(t)) = 0 \ \forall \ t\) where \(\displaystyle y(t) = t^{35} + 4 t^4 - 3 e^{\sin(t)}, x(t) = 2t^{10} - t \tan(t) - e^t\cos(\ln(t))\).

Note that the set of points in the plane via "Cartesian Equation" is given by a function inverse at a point (it's the set \(\displaystyle F^{-1}(0)\)), while in the parametric case it's the image set of a function (the function Q).

Also, if you don't put some minimal restrictions - especially continuity - on the possible functions F and Q for "Cartesian Equations" and parametric equations respectively, then ANY subset of the plane comes from such an F or Q.

Ex: For F, you could use 1 minus the indicator function, meaning F of a point is 0 if the point is in the given set, and 1 otherwise. For Q, one could exploit a set bijection between \(\displaystyle \mathbb{R}\) and \(\displaystyle \mathbb{R}^2\).

Even requiring continuity, there are very counter-intuitive examples possible under these definitions.

Ex: F(x,y) := 0 is the "Cartesian Equation" whose graph is the entire plane. Likewise, parametricly, there are

__continuous__ space-filling curves (like the Peano curve:

https://en.wikipedia.org/wiki/Peano_curve)... counter-intuitive indeed!

A set of points in a plane is not a function! A corresponding function is uniquely recoverable from that only in the first case, where you're told that the set of points is the graph of a function. In the other two cases, "Cartesian coordinates" and parametric equations, there are multiple functions F and Q, respectively, which could produce those sets.

Ex: For the "Cartesian Equation" case, if F defines the set in the plane, then kF defines the same set for any non-zero constant k. For parametric, consider the line y = x via \(\displaystyle x(t) =t, y(t) = t\), OR \(\displaystyle x(t) =t^3, y(t) = t^3\).

The functions F and Q contain more information than you get from just the set of points in the plane given by \(\displaystyle F^{-1}(0)\) and \(\displaystyle \text{Image}(Q)\) respectively!

The "best" form in general is the parametric form, because it contains the most information, is the most flexible, is the most direct (it's just a function \(\displaystyle \mathbb{R} \rightarrow \mathbb{R}^2\)) and is the most useful for modelling.

Parametric Equations common use of t is, of course, for "time". So you think of parametric equations as describing a point moving in the plane in time - which clearly contains vastly more information than just the set that moving point eventually covers. And even more valuable than all that additional information is that the description of a point moving through time is EXACTLY what's needed to model most dynamic real world scenarios.

At the level of generality of your question - asking for "the relationship" between them - that's about all I can think to say.