In geometry (specifically algebraic geometry, such as Shafarevich's Basic Algebraic Geometry), when one refers to a line in affine or projective space, do they mean a straight line? For example, if one considers three points in $\mathbb{P}^2$ that do not lie on a line, what would one mean by a line here?
Secondary question, based on what we mean by a line, given two distinct points in $\mathbb{P}^N$, is there a unique line through these two points in $\mathbb{P}^N$?
Let $(V, +, \cdot)$ be a finite-dimensional vector space over a field $K$, and let $\mathbf{P}(V)$ denote the associated projective space, i.e., the space of one-dimensional subspaces of $V$, usually implemented as $V \setminus\{0\}$ modulo the equivalence relation $x \sim kx$ for every $k \neq 0$ in $K$.
A line in $\mathbf{P}(V)$ is the image of a two-dimensional subspace in $V$. Three points are therefore non-collinear if they are represented by non-coplanar (i.e., linearly independent) vectors in $V\setminus\{0\}$.
Given two distinct points in projective space there is a unique line containing both, namely the image of the plane in $V$ spanned by a choice of representatives.