the line at infinity
In a perspective drawing every set of parallel lines runs off to its own vanishing point, and all those vanishing points lie along one horizontal line across the picture — the horizon. The line at infinity is the geometer's exact version of that horizon: the single line on which all the points at infinity sit, one point for each direction of the plane.
Concretely: when we complete the Euclidean plane into the projective plane, we add one point at infinity per direction. Collected together, those new points are not scattered randomly — they form one line, by definition the line at infinity. In homogeneous coordinates (x : y : z) the ordinary points have z not zero, and the line at infinity is precisely the equation z = 0; its points are the (x : y : 0). It behaves like any other line of the projective plane: it meets every ordinary line in exactly one point (namely that line's own point at infinity), and any two of its points still determine it.
Here is the liberating fact. In the projective plane there is nothing special about the line at infinity — it is an ordinary projective line that we merely chose to call 'infinity'. A projective transformation can swap it with any other line, sending faraway infinity-points to nearby ordinary ones (this is exactly what a camera does when it photographs a horizon onto a finite piece of film). So 'parallel' is not a projective idea at all: two lines being parallel just means they meet on the line we happened to label at infinity, and a change of viewpoint can make that meeting point land in plain sight. Affine geometry is recovered precisely by choosing one line to keep as 'the line at infinity' and refusing to move it.
All horizontal lines (slope 0) meet at one infinity-point; all lines of slope 1 meet at another; all vertical lines at a third. These three points — and every other direction's point — are collinear: they lie on the single line at infinity, the equation z = 0 in homogeneous coordinates.
Every direction's vanishing point lies on one line — the horizon of the projective plane.
Do not think of the line at infinity as a 'boundary' or 'edge' you could fall off. The projective plane has no edge; the line at infinity is a perfectly ordinary line in it, distinguished only by a label we are free to reassign.