Background: Let $\{a_i\}_{i=1}^n$ be i.i.d. random variables with zero-mean and unit variance. Define $$s_n=\frac{1}{\sqrt{n}}\sum_{i\leq n} a_i$$
Central limit theorem says that the distribution of $s_n$ converges to the standard normal ${\cal{N}}(0,1)$.
Short question: For each $n$, give me a Gaussian random variable $g_n$ that is close to $s_n$.
Rigorous question: Fix the number $n$. Can we construct a Gaussian ${\cal{N}}(0,1)$ distribution from $s_n$'s distribution so that if $(s_n,g_n)$ is sampled from the joint distribution we constructed (where $g_n$ is the Gaussian variable) $g_n,s_n$ are close in some sense. For instance, for some $\alpha>0$, we want $${\mathbb{E}}[(g_n-s_n)^2]={\cal{O}}(n^{-\alpha})$$
You are basically asking about a central limit theorem with respect to the Wasserstein 2 metric (you can read more about it here https://en.m.wikipedia.org/wiki/Wasserstein_metric).
Formally, it is equivalent to the classical CLT but it requires more finness to actually find the coupling. You can see some recent results here https://arxiv.org/abs/1506.06966.