You say that "functions look like glorified equations". There's definitely some truth to that. Here's an equation:
$$y = x^2$$
Here's another equation:
$$f(x) = x^2$$
Both of these are equations, and the two equations do very similar things. The two equations both define functions. But they do it a little bit differently.
The first equation doesn't give a name to the function it defines; it just defines $y$ as being "a function of" $x$, which is to say that it tells you what $y$ is once you know what $x$ is. The second equation does give a name to the function it defines; the function is called $f$. (The function isn't $f(x)$; the function is just $f$.)
Now, here's another equation:
$$x^2 + y^2 = 1$$
Unlike $y = x^2$, this equation does not define a function. Why not? Because it doesn't tell you what $y$ is once you know what $x$ is. This equation defines a relation, which is to say that it tells you what the values of $x$ and $y$ are allowed to be, but the value of $y$ is not completely determined by the value of $x$.
A function, it turns out, is just a special kind of relation. A function is any relation that has the property that once you know what the first value is, you know what the second value is. A circle can be described by a relation (which is what we just did: $x^2 + y^2 = 1$ is an equation which describes a relation which in turn describes a circle), but this relation is not a function, because the $y$ value is not completely determined by the $x$ value.
Now, could we use something similar to function notation in order to define a circle? Sure. What we can't do is something like this:
$$x^2 + f(x)^2 = 1$$
Since we're using function notation here, it looks like we're still trying to define a function. But what we can do is give our relation a name. Let's call it $\diamond$. Now we can say this:
$$x \diamond y \text{ whenever } x^2 + y^2 = 1$$
Now, much like $f$ is the name of a function defining a parabola way above, $\diamond$ is the name of a relation defining a circle.