What is a projective plane? How is it different from an affine plane?

I came across the definition in the book titled Elliptic Curves by Anthony W Knapp, couldn't understand it so looked online, which just confused me more. I'm looking for an explanation in the context of curves in projective plane/space.

• In affine geometry you define your space by an equivalence relation $x\sim x+b$ in projective gemoetry instead we take $x\sim ax$ – tired Jun 29 '16 at 8:39
• How do we define space using an equivalence relation? That's what's unclear to me. – DpS Jun 29 '16 at 8:49
• If the concept of "projective space" is not familiar to you, then please allow me to suggest that "elliptic curves" is a subject too complicated for you to study at your present level of mathematical understanding. – Alex M. Jul 6 '16 at 10:09
• Can you suggest me a good book or article or anything to read up projective space first then? – DpS Jul 6 '16 at 10:16

Expressed in coordinates, you add one coordinate. So a point $(x,y)$ in the usual Cartesian plane would be represented as $[x,y,1]$ or any multiple thereof. That's called a homogeneous coordinate vector. So in fact you are no longer dealing in vectors, but strictly speaking in equivalence classes of vectors. Most of the time authors will use the same notation for vectors and for equivalence classes, and rely on context to tell you which is which in those cases where it makes a difference. To convert back, a homogeneous coordinate vector $[x,y,z]$ corresponds to a Cartesian vector $(x/z,y/z)$. If $z=0$, this would be undefined; those are the points at infinity. The vector $[0,0,0]$ has to be excluded, since it would otherwise belong to all equivalence classes. The null vector does not represent any point in the projective plane.
In terms of curves, you might want to make certain that all your equations are homogeneous, i.e. have the same degree in each term (for each geometric object). So for example the equation $7x^2 + 5y = 3$ would not be homogeneous: if you have a vector which solves this, then take twice that vector, you may end up with another representative of the same point which fails the equation. $7x^2 + 5yz = 3z^2$ on the other hand would be a homogeneous equation.