discrete signals - What is meant by *sampling* in terms of the *sampling theorem*?

Let $y:\left(-\frac T2,\frac T2\right)\to\mathbb{C}$ be a square integrable function. The Fourier coefficients of $y$ are $$\underline{Y}(k):=\frac 1T\int_{-T/2}^{T/2}y(t)e^{-i\omega_kt}\;dt\;\;\;\text{with }\omega_k:=k\frac{2\pi}T$$ for $k\in\mathbb{Z}$. The Fourier polynom of degree $n\in\mathbb{N}$ of $y$ is $$\mathcal{F}^{-1}_n[y](t):=\sum_{k=-n}^n\underline{Y}(k)e^{i\omega_kt}$$ and $$\mathcal{F}^{-1}[y]:=\lim_{n\to\infty}\mathcal{F}_n^{-1}[x]$$ is called inverse Fourier transformation of $y$. Now, I've got two questions:

1. What is meant by sampling (in terms of the sampling theorem)? From my understanding, if we know the period $T$ all we need to "store" are the values $\underline{Y}(k)$. We cannot store all values, so we need to choose a "huge enough" $n$ and store only the values $\underline{Y}(-n),\cdots,\underline{Y}(n)$. So, where does "sampling" come into play? The only thing I could imagine is numerical integration: We consider an equidistant grid $$x_j=\left(\frac jN-1\right)\frac T2\;\;\;\text{for }j=0,\ldots,2N$$ and apprximate $\underline{Y}(k)$ using the composite trapezoidal rule, i.e. $$\underline{Y}(k)\approx\frac{1}{2N}\sum_{j=0}^{2N-1}y\left(x_j\right)e^{-i\omega_kx_j}$$ By doing so, we didn't take the whole "signal" $y$, but only the "sample points"$\left(x_j,y\left(x_j\right)\right)$ into account. Is this meant by "sampling"?


2. Does the sampling theorem make a statement about $n$ or $N$ or something else?