First of all why was there a need to
invent projective space?
One place it appears naturally is when we think about
perspective.
Imagine railroad tracks stretching out in front of
you, if you look out to
infinity, at the horizon, the two
lines
seem to meet. We know that parallel lines don't meet in ordinary
Euclidean space but by going to projective space we can make them meet
in a formal
mathematical sense.
This is how it's done. Start with a field k (think of
k=R, the real numbers or k=C,
the complex numbers).
Now for a nonnegative integer n we are going to define
projective nspace. The construction starts with
k^{n+1}\{0}, that is,
n+1tuples of elements of k with the origin 0=(0,0,...,0)
excluded.
We define an equivalence relation on the nonzero points of
k^{n+1} by defining
(a_{0},a_{1},....,a_{n+1}) equiv (b.a_{0},b.a_{1},...,b.a_{n+1})
for any nonzero scalar b in k.
Projective nspace denoted by P^{n}(k)
is the set of equivalence classes of the nonzero points
k^{n+1} under this relation.
It's easier to deal with this once we have some good notation. So we write
[a_{0},....,a_{n+1}] for the equivalence
class of (a_{0},....,a_{n+1}). Thus we have that
[a_{0},....,a_{n+1}]=[b.a_{0},b.a_{1},...,b.a_{n+1}]
Let's do some examples.

n=0 Here P^{0}(k) consists of
the points [a], for a nonzero. But we know that
[a]=[b.a], for any nonzero b. So
P^{0}(k) just consists of one point [1]

n=1 This time P^{1}(k), the projective line,
consists of the points [a,b], for a,b not both zero.
If a=0 then [0,b]=[0,1], so there is only
one point with the first coordinate zero. Suppose on the other hand that
a is nonzero. Then [a,b]=[1,b/a] and so
we see that P^{1}(k) consists of the the points
[1,a] with a in k (i.e a copy
of the usual 1space) together with an extra
point at infinity, namely [0,1].

n=2 Finally P^{2}(k), the projective plane,
consists of the points [a,b,c], for a,b,c not all zero.
This time if a=0 we get a family of points, namely all
[0,b,c] with b,c not both zero. This is just a
copy of the projective line that we have just seen. If a is nonzero
then we get all points of the form [1,b,c]. This is just
a copy of ordinary 2space k^{2}. So the projective plane
is 2space together with a projective line at infinity.
In general, projective
nspace is ordinary
nspace together
with projective
n1space at infinity.