Coplanar
Coplanar points or lines are those that all lie within the same flat plane.
Definition
Coplanar means "in the same plane": points or lines are coplanar if they all fit on the same flat surface, like all being on one sheet of paper (any three points are always coplanar, so it takes four or more to possibly be non-coplanar). Formally, points $P_1, \ldots, P_n$ in $\mathbb{R}^3$ are coplanar if and only if the vectors $(P_2-P_1), (P_3-P_1), (P_4-P_1)$ are linearly dependent, i.e. the $3 \times 3$ determinant $\det([P_2-P_1, P_3-P_1, P_4-P_1]) = 0$, equivalently the scalar triple product $(P_2-P_1) \cdot [(P_3-P_1) \times (P_4-P_1)] = 0$, or more generally the rank of the matrix of displacement vectors is at most $2$.
Example
All the corners of a rectangle are coplanar, but a point floating above the paper is not coplanar with the others. The four vertices of a square are coplanar, while the four vertices of a tetrahedron (triangular pyramid) are not, though any three of its vertices are always coplanar.
Key Insight
Coplanarity matters in 3D geometry: two coplanar lines either intersect or are parallel, while two lines that are not coplanar are called skew lines, they neither intersect nor are parallel. Non-coplanar points define the simplest 3D figure, the tetrahedron, and in linear algebra coplanarity is equivalent to linear dependence of displacement vectors, directly linking this geometric idea to the rank of a matrix.