A scheme (over a field $k$, say) really has two sorts of points and much confusion arises from the fact that they are not distinguished linguistically.
For clarity's sake I'll call them (just here and now!) physical and functorial points.
The physical points They are elements of the underlying set $|X|$. Such an $x\in |X|$ has a residual field $\kappa (x)$ which is an extension $k \to \kappa (x) $. If that extension is an isomorphism, we say that $x$ is rational or $k$-rational.
The functorial points They are $k$ -morphisms from some $k$-scheme $Y$ to $X$.
You are interested in the special case where $Y$ corresponds to a fixed algebraic closure $k\to \bar k$. In that special case, a $\bar k$-point $f:Spec (\bar k) \to X$ of $X$ certainly has an image $x=f(\ast)\in X$.
However the crucial point is that this image does not determine $f$ at all. You also have to give yourself a $k$-algebra morphism $\kappa (x) \to \bar k$ in order to define $f$.
So the same $x$ can correspond to billions of $\bar k$-points, say $7$ billion.
An example Consider $k=\mathbb Q$ and $X=Spec( \mathbb Q[T]/\langle T^{7,000,000,000}-2\rangle)=Spec(K)=\lbrace x\rbrace $.
Although $X$ has only one physical point, namely $x$, there are 7 billion different $\bar {\mathbb Q}$- points in $X$ .
[They correspond -via the affine scheme/ring dictionary- to the $\mathbb Q$-algebra morphisms $K \to \bar {\mathbb Q}$, which in turn are uniquely determined by the choice of a 7,000,000,000-th root of 2 in $\bar {\mathbb Q}$]
The case of varieties In the case of a variety $X$, the closed physical points are exactly the images of the $\bar k$-points of $X$ (see Keenan's answer). Completeness of $X$ is irrelevant.