An elliptical distribution is affinely equivalent to a spherically symmetric distribution. It therefore suffices to specify the radial distribution. By convention, it is given in terms of the distribution of the squared radius. When that is continuous, its density is the generator.
This means that when an $n$-dimensional random variable $X$ has an elliptical distribution, there is a center $\mu\in\mathbb{R}^n$ and an invertible matrix $\Lambda$ such that the distribution of
$$Z = \Lambda^{-1}(X-\mu)$$
is the same as the distribution of $QZ$ for any orthogonal matrix $Q.$ Equivalently, the distribution of $Z$ determines and is determined by the distribution of its squared Euclidean length
$$|Z|^2 = Z^\prime Z = (X-\mu)^\prime \Lambda^{-1^\prime} \,\Lambda^{-1}(X-\mu) = (X-\mu)^\prime \Sigma^{-1} (X-\mu)$$
where $\Sigma=\Lambda \Lambda^\prime$ is (clearly) symmetric and positive definite.
When such a distribution is continuous it has a multivariate density $f,$ which therefore can be written in the form
$$f(x) = k\,g\left((x-\mu)^\prime \Sigma^{-1} (x-\mu)\right)$$
with $k$ the constant that makes $f$ integrate to unity. According to Wikipedia, $g$ is a generator of the distribution of $X.$
For example, because the standard elliptical Normal distribution has a density proportional to $$\exp(-(z_1^2+z_2^2+\cdots+z_n^2)/2) = \exp\left(-|z|^2/2\right) = g(|z|^2),$$ its generator is proportional to
$$g(z) = e^{-z/2}, z \ge 0;\ g(z) = 0, z \lt 0.$$
(It's always understood that $g$ only needs to be defined for $z\ge 0.$)
Note that $g$ is not the density of $|Z|^2$ nor even of $|Z|$ itself: that's because in spherical coordinates in $\mathbb{R}^n,$ $g$ must be multiplied by $|z|^{n/2-1}$ to obtain the density of $|Z|^2.$ For instance, the distribution of the squared spherical Normal radius has a density proportional to $z^{n/2-1}e^{-z/2},$ a Chi-squared distribution.
Because it can help to have more than one example of something, here are a few other generators.
The multivariate Student t distribution
For the distribution with $\nu$ degrees of freedom, a generator is
$$g_\nu(z) = \frac{1}{\left(1 + z\right)^{(\nu+n)/2}}.$$
The multivariate uniform spherical distribution
A generator is
$$g(z) = 1, 0 \le z\le 1;\ g(z) = 0\text{ otherwise.}$$
This concept of "generator" has little to do with copulas per se and almost nothing to do with tail behavior, either. ($g$ determines its tail behavior, of course; but there are many possible functions $g$ associated with any radial tail behavior.)
The account I gave of the Mahalanobis distance applies equally well to understanding any multivariate distribution, showing how $\mu$ merely establishes the origin of a coordinate system and $\Sigma$ determines a set of orthogonal coordinate axes in which the multivariate distribution of $X$ has a particularly simple form. What makes the distribution "elliptical" is that the resulting form is especially simple, depending only on the (Mahalanobis) distance from the origin.
