Collinear
Collinear points are three or more points that all lie on the same straight line.
Definition
Collinear means "on the same line": if three or more points are collinear, one straight line passes through all of them (any two points are always collinear, since two points determine a line, so the interesting case is three or more). Formally, points $P_1, P_2, \ldots, P_n$ in $\mathbb{R}^n$ are collinear if the vectors $(P_2 - P_1), (P_3 - P_1), \ldots, (P_n - P_1)$ are all scalar multiples of a single vector, equivalently if the matrix formed by $(P_i - P_1)$ has rank at most $1$.
Example
If you place three dots on a ruler and they all touch its edge, those dots are collinear. Points $(0, 0)$, $(1, 2)$, and $(2, 4)$ are collinear because they all satisfy $y = 2x$, while $(0, 0)$, $(1, 1)$, and $(2, 3)$ are not, since no single line passes through all three. Three points $A, B, C$ in $\mathbb{R}^2$ are collinear if and only if $\det([B-A, C-A]) = 0$, that is, $(B_x-A_x)(C_y-A_y) - (B_y-A_y)(C_x-A_x) = 0$, a determinant that equals twice the signed area of triangle $ABC$.
Key Insight
The word "collinear" comes from Latin, "co-" meaning together and "linear" meaning line, so it simply means "together on a line"; non-collinear points are needed to define a triangle or a plane. In projective geometry, collinearity is a projective invariant, projections and perspectivities map collinear points to collinear points, which is the foundation of cross-ratio theory and the projective transformations used in computer vision.