TITLE: Motivating Example for Algebraic Geometry/Scheme Theory
QUESTION [39 upvotes]: I am in the process of trying to learn algebraic geometry via schemes and am wondering if there are simple motivating examples of why you would want to consider these structures.  
I think my biggest issue is the following: I understand (and really like) the idea of passing from a space to functions on a space.  In passing from $k^n$ to $R:=k[x_1,\ldots,x_n]$, we may recover the points by looking at the maximal ideas of $R$.  But why consider $\operatorname{Spec} R$ instead of $\operatorname{MaxSpec} R$?  Why is it helpful to have non-closed points that don't have an analog to points in $k^n$?  On a wikipedia article, it mentioned that the Italian school used a (vague) notion of a generic point to prove things.  Is there a (relatively) simple example where we can see the utility of non-closed points?

REPLY [3 votes]: A very late answer, my apologies if I am not contributing anything that the very wonderful answers above have already contributed.
Generic points $\Leftrightarrow$ irreducible subsets... as long as we are doing classical geometry, there really is no major difference between using $\text{Spec}$ and using $\text{MaxSpec}$ and talking about things "being true outside a closed subset." And this is just an incredibly useful thing to do; for example, if we want to do an induction by taking hyperplane sections, we can talk about a randomly-chosen hyperplane as a generic hyperplane, and this does not really cause any issues.
In fact, in Mumford's Algebraic Geometry I: Complex Projective Varieties, Mumford literally defines a generic point as a point with coefficients transcendentally independent from the coordinates of our equations, and thinking in this way causes zero hinderance as long we are classical.
The real power of generic points is as an extremely convenient language for all these things, which in particular, is infinitely cleaner once we are not working over a field like $\mathbb{C}$ which is absurdly large. For example, let us say that we want to prove that a variety of dimension $n$ is generically smooth. The point is that we can define "smooth at a point" in a way that makes sense for "normal points" and generic points, so we can literally check that the generic point is smooth by plugging in equations. If one thinks about this, we are basically "formally taking a point with transcendental coefficients."