The basic construction, given a vector space V over a field K, is to form the set of equivalence classes of non-zero vectors in V under the relation of scalar proportionality: we consider v to be proportional to w if v = cw with c in K non-zero. This idea goes back to mathematical descriptions of perspective. If the field K is the real numbers, and V has dimension n, then the projective space P(V) - which we can talk about as the space of lines through the zero element 0 of V - carries a natural structure of a compact smooth manifold of dimension n-1. It is also highly symmetric, since any linear automorphism of V gives rise to a symmetry of P(V). These in the classical examples identify with 'perspectivity' and 'projectivity' transformations described geometrically, and account for the name. The group of these symmetries is the quotient of the general linear group of V by the subgroup of non-zero scalar multiples of the identity.
The use of projective spaces makes quite rigorous the talk about a 'line at infinity' (where parallel lines meet), or a 'plane at infinity' for three dimensions: a translation of the latter can be made as part of the projective space associated to a four-dimensional real vector space. In that way geometrical ideas introduced by Poncelet and others become part of a theory founded on linear algebra. The part of a projective space not 'at infinity' is called affine space; but the symmetries of P(V) do not respect that division. Use of a basis of V allows, if required, the introduction of homogeneous co-ordinates for the handling of concrete calculations.
Use of vector spaces over the field of complex numbers gives rise to different manifolds, also used by geometers. There are good reasons for using them, in order to get a theory about intersections of algebraic varieties with predictable properties. In the theory of Alexander Grothendieck there are reasons for applying the construction outlined above rather to the dual vector space V'.