... variables¹

There are a couple of subtleties here. Usually we will insist that $f(x,y)$ be non-constant, since if $f(x,y)$ is a constant, then the set of solutions to $f(x,y)=0$ is either empty or the entire plane, depending on whether the constant is nonzero or zero. Also, although we will usually draw the set of solutions to $f(x,y)=0$ where $x$ and $y$ are real numbers, the theory actually works better when one allows complex number solutions as well. For example, the ``curve'' $x^2+y^2+1=0$ looks empty if one only takes real number solutions, but acquires many solutions if $x$ and $y$ are permitted to be complex numbers.

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...conics²

This is because they arise by slicing a double cone in space such as $x^2+y^2=z^2$ with a plane.

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... nonsingular³

We have not defined this term, but loosely speaking it means that $X$ has no ``corners'' or points where the curve ``crosses itself.''

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