TITLE: Why are projective spaces and varieties preferable?
QUESTION [8 upvotes]: I am reading Hartshorne's Algebraic Geometry and it seems to me that projective spaces and varieties are prefferable. I don't know why. 
In a more elementary stage of mathematics, when we try to find solutions to given equations, in $\mathbb R$ or $\mathbb C$, are we thinking in a more affine way? I really find it easier to understand things or to draw pictures in affine spaces, since it "looks" much more like the space we are living in. 
So why do people, for example Hartshorne, always transform problems in affine spaces to problems in projective spaces? Why do they prefer projective spaces and varieties? What books are more basic and thorough on picturing varieties in projective spaces?

REPLY [7 votes]: Looking at problems in the projective setting is a kind of compactification. Compactness is a strong property which makes life easy lots of the time. 
I will make this a community wiki answer and start a list of statements that make working with projective varieties particularly interesting:

The image of a morphism $f:X\to Y$ is closed if $X$ is projective. In general, the image of a morphism of varieties is not a variety.
As N.H. already mentioned, intersections in projective space behave "well" in the sense of Bézout's Theorem.