From a "physical" point of view, this means that we are only interested in the variation of $S$ as $E$ changes, even though probably changing $E$ might also (if $N$ depends on $E$ in some way) somewhat affect $N$ and therefore change $S$ not directly by the change of $E$ but rather through $N$. The partial derivative you wrote ignores this "implicit", "hidden" effect. What follows is a more detailed explanation.
The partial derivative you wrote, ${\partial S \over \partial E}|_N$, means you can treat $N$ as a constant. So for example if
$$S=\alpha N^2E$$ its partial derivative with respect to to $E$ with $N$ constant is simply $\alpha N^2 $ because $N^2$ can be treated as a constant (I just used a random expression, there is no entropy I can think of with that expression..!).
This is usually not an issue when doing computations, you just regard $N$ as a constant and that's it, but it does avoid confusion when both $N$ and $E$ are functions of another parameter, say $w$ so that $E=E(w)$ and $N=N(w)$. In this case, you cannot claim to be just "slightly" changing $E$ keeping $N$ constant because they are intertwined by $w$. In this case, if what interested you is the total rate of change of $S$ then you would have to compute the total derivative of $S$ i.e. (by the chain rule):
$${dS(E(w), N(w)) \over dw}={\partial S \over \partial E}|_N{dE\over dw}+{\partial S \over \partial N}|_E{dN\over dw}$$
You see that this expression takes into account not only the variation of $S$ with respect to $N$ but also the contribution due to the variation of $E$.
Also, in general, it could be that $N$ is a function of $E$, so that for example:
$$S=\alpha N(E)^2E$$
In this case, when computing the partial derivative at constant $N$ we ignore this dependency and we are only concerned in the variation of $S$ as the part of it that explicitly contains $E$ varies, so that again we get $${\partial S \over \partial E}|_N=\alpha N(E)^2$$ because we treat $N$ as constant even though it contains an implicit dependency on $C$. Differently the total derivative would be (by the chain rule / by the formula above with $E$ instead of $w$):
$$dS/dE=\alpha N(E)^2+ 2\alpha N(E){dN\over dE} E$$