Short answers, pulled from the story.
In geometry, a straight line is an infinitely long object with no width, depth, or curvature. It is a space of dimension one and a special case of a curve, and it idealizes physical objects such as a straightedge, a taut string, or a ray of light.
Euclid's Elements defines a straight line as a breadthless length that lies evenly with respect to the points on itself. This definition relies on undefined terms and is never used in the proofs of theorems in the rest of the text.
On a Euclidean plane, parallel lines are lines in the same plane that never cross. Skew lines exist in three-dimensional space, are not in the same plane, and therefore do not intersect each other.
Three or more points are collinear when they lie on the same line. In a plane, points are collinear if and only if the slope between one pair of points equals the slope between any other pair, or equivalently if a corresponding matrix has rank less than three.
The Hesse normal form, named after the German mathematician Ludwig Otto Hesse, is based on the normal segment drawn from the origin perpendicular to the line. Unlike the slope-intercept and intercept forms, it can represent any line while requiring only two finite parameters.
In the spherical representation of elliptic geometry, lines are represented by great circles of a sphere with diametrically opposite points identified. In a different model, lines are represented by Euclidean planes passing through the origin, and both representations satisfy the property that two points determine a unique line.