What is the use of scheme theory? I should preface this by saying that my background in Algebraic Geometry is (more or less) the content of Vakil's notes up through Chapter 4 (i.e. through the definition of a scheme and several examples, I haven't yet read carefully about Proj). 
I am finding myself unmotivated to move further on in these notes, since I don't so much see the point of scheme theory. I am not stupid enough to think that there isn't a point, so I was wondering if someone could motivate (even briefly) the study of this (admittedly difficult) abstraction. I am only vaguely familiar with the classical case, so perhaps that has something to do with it.
Thanks.
 A: I think that perhaps you're asking a slightly wrong question.  If you don't have any
motivation to learn schemes (e.g. from Vakil's notes), then probably that's not what you should be studying right now.
Instead, try to learn some classical algebraic geometry, such as the basics of the theory of curves.  Miles Reid's Undergraduate algebraic geoemtry is a good place
to start (even if you're not an undergraduate); Silverman's Arithmetic of elliptic curves (the first volume) ultimately veers towards number theory (not that that's a bad thing!), but begins with quit a bit of geometry of curves.
Chapters IV and V of Hartshorne give a beautiful treatment of some of the basics of
curves and surfaces, although from a view-point that is slightly unforgiving for a novice (even if you're willing to take the earlier foundational material on faith).
In any case, if you look at a few such texts, you can see whether you actually
like algebraic geometry, and also (if you do) whether your taste lies more towards
geometry proper, or more towards arithmetic geometry/number theory.
At that point, as you begin to pursue your inclinations in more depth, you will
naturally find yourself needing to learn scheme-theoretic foundations, and hopefully will have the motivation to do so.

Okay, after that rant, here is a more literal answer to your question:
firstly, schemes are not necessary for the study of algebraic geometry (there
are plenty of excellent geometers with a more analytic bent, who use complex analytic, and related, techniques, rather than schemes), but they form one of
the basic approaches to the modern theory, and are particularly indispensable
in arithmetic geometry (the part of algebraic geometry that overlaps with number 
theory; basically it refers to the study by algebraic geometry methods of 
Diophantine equations).
An affine algebraic variety is basically something cut out by some equations $f_1 = \cdots f_r = 0$ in some variables $x_1,\ldots,x_n$, with coefficients in some field $k$.  We can encode this in the ring $k[x_1,\ldots,x_n]/(f_1,\ldots,f_r).$
If you worry about the actual solutions to this equation, you start to fuss 
about whether or not this ring has nilpotents (because elements of $k$ can't tell
the difference between the condition $x^2 = 0$ and the simpler condition $x = 0$),
but Grothendieck's idea is just to take the ring itself.
Then, in e.g. number theory applications, we want to replace $k$ by $\mathbb Z$
or the $p$-adic integers.  Or if we have equations depending on parameters,
then the $f_i$ won't have coeffs. just in $k$, but in the ring obtained by
adjoining the parameters to $k$.
This leads to more general rings than just f.g. algebras over fields.
Finally, a key fact in classical alg. geom. is that we can "glue" affine
varieties together to make e.g. projective varieties, because affine varieties
have a topology (the Zariski topology).  
Grothendieck saw how to convert a ring into a space with a topology, a so-called
affine scheme, and then defined schemes to be the things you can get by gluing together affine schemes.  Because of wanting to remember rings themselves,
and not just the points obtained by solving equations in fields, he had to add
a structure sheaf to the data, so schemes are not just top. spaces, but locally ringed spaces.  
As you can see, I mentioned three key ideas as motivation: the possibility of nilpotents (already mentioned by Sasha Patoski), the possibility of working over $\mathbb Z$ in number theory applications, and the possibility of working with parameters.  These are the three big applications of scheme-theoretic ideas,
but to really get the point of them in any detail, you need to know something
about classical algebraic geometry and/or number theory, to get a feeling for
the kind of problems that come up and that are resolved by scheme-theoretic 
arguments/techniques.
A: One of the possible explanations why considering schemes is important is the following. In some sense, schemes are like varieties, but "with nilpotents". What I mean is the following. 
Suppose you are intersecting two plain curves $y=0$ and $y=x$, and also intersect $y=0$ with $y=x^2$. Set-theoretically, in both cases you get the origin $(0,0)$. But really in any small neighborhood the picture will be different in these two cases. So, considering set-theoretic intersection will lose information (it will not remember where this point came from, i.e. it will forget the infinitesimal information about the intersection). But, if you consider the ideals these equations define (i.e. consider scheme-theoretic intersection), you will get different answers. Indeed, the intersection in the first case will be given by the quotient $k[x,y]/(y,x-y)\simeq k[x]/x\simeq k$. In the second case you will get $k[x,y]/(y,y-x^2)\simeq k[x]/(x^2)$. So the scheme "remembers" where the intersection of the curves (the origin) came from. 
This example is not very far from general case. Indeed, whenever you have a scheme $X$, you can obtain another scheme $X^{red}$, closed points of which will give you variety. A very nice introduction into schemes which gives a lot of examples and motivation (and which also explains what $X^{red}$ means) is the book by Eisenbud-Harris called "Geometry of schemes".
