projective space
Imagine standing at the origin and looking out: every line of sight through the origin is one 'direction'. Projective space is the space of all such directions. Two points of ordinary space that lie on the same ray through the origin become a single point here, and this small change tidies up geometry enormously, because directions that are 'parallel' in the affine world now genuinely meet.
Formally, projective n-space P^n over a field k is the set of lines through the origin in the vector space k^{n+1}, equivalently the quotient of k^{n+1} minus the origin by the relation v ~ tv for t a nonzero scalar. A point is written in homogeneous coordinates (x_0 : ... : x_n). It is covered by n+1 standard affine charts U_i where x_i is nonzero, each isomorphic to A^n, and these glue to give P^n the structure of a smooth projective variety.
Projective space is the natural home of projective varieties and the basic compactification of affine space: A^n embeds as the chart x_0 nonzero, and the leftover hyperplane x_0 = 0 is a copy of P^{n-1}, the 'points at infinity'. This completeness is what makes Bezout's theorem clean — two plane curves of degrees d and e meet in exactly de points over an algebraically closed field, counted with multiplicity, once one works in P^2.
P^1 over C is the Riemann sphere: the affine line A^1 with coordinate x = (x : 1), plus the single point at infinity (1 : 0). Two distinct lines in P^2 always meet in exactly one point.
Compactifying the line into a sphere is the prototype of the projective construction.