the projective plane
Look down a long straight road or a pair of railway tracks: the two parallel rails seem to meet at a single point on the horizon, even though, in plain Euclidean geometry, parallel lines never meet. The projective plane is the geometer's honest answer to that everyday illusion. It is the ordinary plane with just enough extra points added — one for each direction — so that every pair of lines, parallel or not, meets in exactly one point. No more exceptions, no more 'unless they happen to be parallel'.
Build it like this. Start with the familiar Euclidean plane. To every family of parallel lines (all the lines sharing one slope) attach a single brand-new point, called a point at infinity, where they all agree to meet. Different slopes get different points at infinity; all these new points together form one extra line, the line at infinity. The result — old plane plus the line at infinity — is the real projective plane, written RP^2. In it two distinct points always determine exactly one line, and two distinct lines always meet in exactly one point: the two basic facts now stand perfectly symmetric. There is also a beautiful concrete model: take ordinary three-dimensional space and let each 'point' of the projective plane be a whole line through the origin, and each 'line' be a whole plane through the origin. A line-through-origin meets the drawing-plane z = 1 in a single ordinary point, unless it is horizontal, in which case it is one of the points at infinity.
Why bother? Because dropping the parallel exception makes theorems shorter, removes dozens of special cases, and reveals a deep symmetry (duality) between points and lines that Euclidean geometry hides. It is honest to stress what the projective plane is not: it has no notion of distance, no angles, no 'betweenness', no parallel/non-parallel distinction at all — those belong to finer geometries built on top of it. It keeps only what survives projection (shadows, perspective drawing, a camera's view): which points lie on which lines. That spareness is exactly its power.
Take the two parallel lines y = 0 and y = 1 (slope 0). In the Euclidean plane they never meet. In the projective plane they share one point at infinity — the common 'horizontal direction' point — so they meet there, exactly once. The line y = 5x and the line y = 5x + 3 (slope 5) share a different point at infinity. All such infinity-points lie on the single line at infinity.
Adding one point per direction turns 'parallel lines never meet' into 'every two lines meet exactly once'.
A common misconception: there is not one single 'point at infinity' but a whole line of them — one for each direction. And the points at infinity are not 'special' or 'less real' than ordinary points; in the projective plane every point is on an equal footing, and a suitable transformation can send any infinity-point to an ordinary location.