Let $X$ be a scheme over an algebraically closed field. We say that $\Gamma(X,\mathcal{O}_X)$ separates points iff for every $x,y \in X$ there's an $f \in \Gamma(X,\mathcal{O}_X)$ with $f(x)\ne f(y)$.
This is true for affine schemes. Beyond that it seems a very restrictive property. Does it have any interesting uses? Is there an alternative charactrization for such schemes?
The definition: danger ahead!
The global functions of a scheme $X$ are said to separate points if, given two points of $ X$, there is a function $f\in \Gamma(X,\mathcal O_X)$ which is zero at one of them and non-zero at the other.
Beware that we may NOT choose the one at which it is zero: for example if $X$ is an integral scheme with generic point $x$, any function zero at $x$ will vanish at all other points and in particular at $y$. However it is still possible for a function to be zero at $y$ and non-zero at $x$.
(Recall that $f$ is zero at $z$ if $f[z]=f_z\operatorname {mod} \mathfrak m _z =0\in \mathcal O_{X,z}/\mathfrak m_z$ . See here)
Quasi-affine schemes
A vast supply of schemes whose global regular functions separate points is provided by the quasi-affine schemes: those are the open subsets of affine schemes.
Here is a criterion for a scheme $X$ to be quasi-affine :
$$ \text {The scheme} \:X \: \text {is quasi-affine} \\ \iff\\ \operatorname {The canonical scheme morphism} j: X\to \operatorname {Spec} (\Gamma(X,\mathcal O_X)) \operatorname {is an open embedding} $$ A notorious example:
The scheme $\mathbb A^2\setminus \{O\}$ is quasi-affine but not affine.
A close analogy
In complex analysis, spaces whose global functions separate points are called holomorphically separable. They are an important ingredient in characterizing Stein spaces.
I'm not aware that there is a similar terminology in algebraic geometry: I find this regrettable [the lack of terminology or my ignorance of it :-)]
Their exact opposite
Complete connected varieties $X$ over an algebraically closed field $k$ (for example projective varieties) have the property that $\Gamma(X,\mathcal O_X)=k$.
So as soon as they are positive dimensional two points cannot be separated by global functions (but they can be separated by sections of appropriate line bundles) .
What are they good for?
Look here: a quasi-affine variety is confronted to a complete variety to prove that two varieties must intersect in projective space.
What else is there?
A paraphrase of the definition at the beginning is: $X$ is quasi-affine iff the canonical scheme morphism $j: X\to \operatorname {Spec} (\Gamma(X,\mathcal O_X))$ is injective.
This is quite close to the criterion above for $X$ to be quasi-affine, so that my intuition is: quasi-affine schemes constitute the "majority" of schemes whose global functions separate points.
