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I am confused about what are the points of the topological space $|X|$ underlying a scheme $X$. When $X = \textrm{Spec}(A)$ is affine, the points of the topological space are just the prime ideals of $A$.

The cause of the confusion is the idea of geometric points, which are morphisms from Spec of an algebraically closed field to $X$. I don't think these are points of the underlying topological space $|X|$ in general. So my question is: can the points of $|X|$ be described in terms of the functor of points approach? (I mean the points of $|X|$ correspond to morphisms $\textrm{Spec}(K) \to X$ for what fields $K$?)

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    A good answer relies on your definition of $|X|$. Can you be more precise- are generic points included in $|X|$, for instance? – 2017-01-24

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A scheme is "locally affine". That means if I take any point $p\in X$, there is an affine open neighborhood $U$ containing $p$, say $(U,\mathscr O_X|_U)\cong(Spec(A),\mathscr O_{Spec(A)})$. Therefore $p$ corresponds to some prime ideal $\mathfrak p\subset A$. This is often the way you work with the points in a "hands on" way while doing problems. So for instance, the stalk at $p$ is $\mathscr O_{X,p}\cong A_{\mathfrak p}$.

Unfortunately, I don't think I have a good response to the second part of your question. Hopefully somebody else will.

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    Thanks for your answer Alex, but the second part of my question was really the question I was asking. I edited the question to hopefully make this more clear. – 2017-01-23
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I don't think such a description is possible:

The fields in question ought to be exactly the residue fields of points in $X=Spec(R)$ (It's enough to consider affine schems here). Clearly any point $x\in X$ has a residue field $K_x$, which gives us a morphism. $Spec(K_x)\to X$.

Similarly, for any morphism $f: R\to K$, $K$ any field, consider $Ker(f)$. Since $Im(f)$ is a subring of $K$, it is an integral domain. Thus $Ker(f)$ is prime $p$. Thus this morphism determines a point of $X$. This point has residue field a subfield $K'$ of $K$, containing $im(f)$. So we can think of $p$ being determined by the induced map $f: R\to K'$. (The associated map $R_p\to K'$ will be surjective with kernel $p$).

So geometric points correspond to $\it{extensions}$ of residue fields of points in $X$, so there will be many geometric points which determine one regular point.

The subtlety is that if $K_x$ is a residue field at $x$, and $K_y$ is a residue field at $y$, we could have $K_x$ an extension of $K_y$. Thus a map $R\to K_x$ whose image lies in $K_y$ actually corresponds to $y$, when $y$ is already determined by the restriction of this map. So a functor of points description shouldn't be possible.

As an aside, closed points are certainly exactly surjective ring maps $R\to K$, and points in general will be exactly the surjective maps from $R_p\to K$, for $p$ a prime, which isn't very helpful.