For a domain (or integral domain), the affine scheme is called an integral affine scheme. We know that the spectrum of the ring Spec $R$ is a collection of all the prime ideals of the ring, which is equivalent to geometric points of the affine space. For instance, for the polynomial ring $\mathbb{Z}[x]$, the affine space is
$$A^1_{\mathbb{Z}} = Spec(\mathbb{Z}[x]).$$
For an algebraic closed field $K[x_1, x_2, \cdots, x_n]$, the prime ideals are of form $(x_1-a_1, x_2-a_2, \cdots, x_n - a_n )$ where $a_n \in \mathbb{C}$. This is known as weak Nullstellanz. (In a previous case, we saw that for $\mathbb{C}[x]$, the prime ideals were the maximal ideals $(x-a)$ where $a \in \mathbb{C}$ and the zero prime ideal $(0)$.)
We called the prime ideal (0) a `generic point' in that picture. But what does the term mean? Generally, a generic point $x \in X$ where $X$ is a topological space if the closure of $x$ is the whole space. (One can find different definitions of generic points in the presence of different motivations.) Equivalently, we say that the generic point is a point that is `generic' for the whole space. Thus, if some function is valued on the generic point, then the function will value the same everywhere in the space. In general, a generic point is not available in the affine space. So, a generic point is contained in any other point of space. In the example of $\mathbb{C}[x]$, the generic point is unique, which is zero ideal. We know that the affine space points correspond to the ring's prime ideals. But a zero ideal can not be `pointed' in the affine space, meaning that the points of affine space will generally correspond to the maximal ideals of the ring.
The prime ideals of $\mathbb{Z}[x]$ are the principle ideals generated by primes $p$, ideals generated by irreducible polynomials $f(x)$, of form $(p,(f(x))$ which are the maximal ideals and zero ideals. Now, we know that the affine space of $\mathbb{Z}[x]$ is just a space with points corresponding to these prime ideals, which are called the spectrum of $\mathbb{Z}[X]$. Interestingly, Mumford has a picture containing these points in his Red Book of Varieties and Scheme, known as Mumford's Treasure Map.
We see that the map has some points on the intersection of the horizontal and vertical curves, and the curves themselves are a collection of prime ideals. The horizontal curves are the prime ideals generated by some irreducible polynomial of form $f(x)$. In the map, one has polynomial $(x)$, so the ideal is $Z[x]/(x) \simeq \mathbb{Z}$ and similarly we have $(x^2+1)$ and so on. (We can see that the curve of $(x)$ is less thickened than $(x^2+1)$, which is because of the number of elements contained in the ideal.)
The vertical curve has points of the principle ideals the primes generate, for example, $\mathbb{Z}[x]/p$. $\mathbb{Z}[x]/2$ which is just $\mathbb{F}_2[x]$ (since $\mathbb{Z}[x]/p \simeq \mathbb{Z}/(p\mathbb{Z}) [x]$) where $\mathbb{F}_p$ is a finite field. Now, we have the points on the intersection of the curves, which are $(p,(f(x))$, and these are the maximal ideals. So, for $(2,(x+1))$, we have $\mathbb{Z}[x]/(3,(x+2)) = \mathbb{Z}/(3\mathbb{Z})[x]/(x+2) = \mathbb{F}_3$.
But where are the generic points? Mumford has some doodles in the upper right corner of the map. This is the zero ideal of $\mathbb{Z}[x]$ and is called a generic point. Geometrically, it does not make sense to point out a generic point since it is available everywhere, but it is nicely drawn on the map. The doodle has been pointed in every direction and is contained in every other point of the space. Once again, a generic point is quite harder to make sense of geometrically, but this is a nice way of visualizing them for the case of $\mathbb{Z}[x]$.
