= Scheme point
A scheme point is a point of the underlying <topological space> of a <scheme>, together with the local algebra supplied by its <structure sheaf>. On an affine scheme $\operatorname{Spec}R$, points correspond to <prime ideals> $\mathfrak p\subset R$, and the <local ring> at the point is $R_{\mathfrak p}$. Closed points correspond to <maximal ideals>; schemes also have nonclosed points.
For $\operatorname{Spec}\mathbb Z$, each prime number $p$ gives a closed point $(p)$ with <residue field> $\mathbb F_p$. The prime ideal $(0)$ is a nonclosed generic point with residue field $\mathbb Q$. Its closure is the whole spectrum, so it records the common generic arithmetic rather than one reduction modulo a prime.
More generally, the point for $\mathfrak q$ is a specialization of the point for $\mathfrak p$ exactly when $\mathfrak p\subseteq\mathfrak q$. Thus incidence in the prime-ideal ordering becomes topology. Treating every scheme point as an ordinary coordinate tuple would discard this generic and specialization information.
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