What elementary problems can you solve with schemes? I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary self-contained problems that scheme theory answers - ideally something that I could explain to a fellow grad student in another field when they ask "What can you do with schemes?"
Let me give an example of what I'm looking for: In finite group theory, a well known theorem of Burnside's is that a group of order $p^a q^b$ is solvable. It turns out an easy way to prove this theorem is by using fairly basic character theory (a later proof using only 'elementary' group theory is now known, but is much more intricate). Then, if another graduate student asks me "What can you do with character theory?", I can give them this example, even if they don't know what a character is. 
Moreover, the statement of Burnside's theorem doesn't depend on character theory, and so this is also an example of character theory proving something external (e.g. character theory isn't just proving theorems about character theory).
I'm very interested in learning about similar examples from scheme theory. 

What are some elementary problems (ideally not depending on schemes) that have nice proofs using schemes?

Please note that I'm not asking for large-scale justification of scheme theoretic algebraic geometry (e.g. studying the Weil conjectures, etc). The goal is to be able to give some concrete notion of what you can do with schemes to, say, a beginning graduate student or someone not studying algebraic geometry.
 A: It seems to me that there are a lot of great answers here but I am not sure if they live up to the challenge of showing the usefulness of schemes through an elementary example.
Let me try my luck and risk the wrath of the MO crusaders.
Probably anyone having any kind of mathematical background have seen the classification of conics (a.k.a. plane quadrics) over $\mathbb R$. Perhaps still a large percentage of those who saw this early in their mathematical career wondered about the asymmetry involved in that you have the usual nice ones (ellipse, parabola, hyperbola), the reasonable degenerate ones (two different lines, either intersecting or parallel), and then there are the weird ones (double line, point, empty set). 
This last bunch, while clear from the proof seems odd and one might feel that they should not be considered. Now if one looks at them scheme theoretically, then it becomes completely clear how these fit into the same mold and how those "weird" ones are really the same as the nice ones (some points, some empty sets) or the "reasonable" degenerate ones (double lines, other points, other empty sets). The double line especially is hard to explain without schemes, while the other two "weird ones" are really a consequence of $\mathbb R$ not being algebraically closed, although interestingly, the associated schemes actually kind of "see" the non-real points as well.
A more high brow version of the same idea is the fact that (say over $\mathbb C$) a family of elliptic curves (topologically tori) can degenerate to a nodal cubic curve (topologically a sphere with two points glued together) and a family of rational curves (topologically spheres) can also degenerate to the same object. This seems to lead to at least confusion if not to contradiction as it seems to indicate that a sphere can be continuously transformed into a torus. Looking at these families the proper way, that is, using schemes we see that the two degenerations are not producing the same scheme. The one coming from the rational curves will add an embedded point at the node while that embedded point does not appear in the family of elliptic curves.
A: Ellenberg-Venkatesh-Westerland have a very nice result about realization of abelian $p$-groups as the $p$-part of class groups of certain function fields. All this fits in the general program of studying Cohen-Lenstra heuristics over function fields. One key scheme theoretic player in E-V-W result is a certain type of scheme called Hurwitz scheme. 
To read great stuff related to all this check:
Ellenberg-Venkatesh, Statistics of Number Fields and Function Fields (Proceedings of ICM 2010)
Ellenberg-Venkatesh-Westerland http://arxiv.org/pdf/0912.0325
I'm almost sure you've already read this, but just in case
https://quomodocumque.wordpress.com/2009/12/12/homological-stability-for-hurwitz-spaces-and-the-cohen-lenstra-conjecture-over-function-fields/
A: This was my own motivation for learning schemes:
Theorem (Mazur): If $E$ is an elliptic curve over $\mathbb Q$, then the torsion subgroup
of $E(\mathbb Q)$ (the set of rational points of $E$) is isomorphic to $\mathbb Z/n\mathbb Z$ for $n = 1,
\dots, 10,$ or $12$, or $\mathbb Z/2n\mathbb Z \times \mathbb Z/2\mathbb Z$ for $n = 1,\ldots,4$.  
A special case (due to Mazur and Tate) is
Theorem: If $E$ is an elliptic curve over $\mathbb Q$, then $E$ does not contain a rational point of order $13$.
This is certainly a simple statement, but their proof uses the theory of schemes in a crucial way. 
A: If $I,J \subset A$ are comaximal ideals in a commutative ring $A$, i.e. $I+J=A$, then 
 for all $n,m \in \mathbb N$ the ideals  $I^n$ and $J^m$ are also comaximal.
Proof: $\emptyset= V(A)=V(I+J)=V(I)\cap V(J)=V(I^n)\cap V(J^m)=V(I^n+J^m)$ hence $I^n+J^m=A$. 
Warning: I wouldn't like to be drawn into a discussion on whether this is just terminology or trivial algebraically or a big cheat or what not. All I know is that when I had to prove this result a long time ago, I came up with this proof a few months after I had started learning affine schemes and I was exhilarated at the thought that I could literally see why the result held by drawing two disjoint little doodles representing $V(I)$ and $V(J)$ inside a potato representing $Spec(A)$.    
Edit (April 8th, 2016)
Here is an example of how thinking scheme-theoretically led to a proof of a purely algebraic problem.
A: The fact that a first-order statement about algebraically closed fields holds in characteristic zero if and only if it holds in all large enough finite characteristic can be proved using Chevalley's theorem on images of constructible sets, together with a routine cataloguing of the constructible subsets of $\mathrm{Spec}(\mathbb{Z})$.  This gives a bunch of examples, including the Ax-Grothendieck theorem on injective polynomial mappings.
(The idea of the proof is you assign to a free formula in n variables $x_1,\ldots,x_n$ the subset of $\mathrm{Spec}(\mathbb{Z}[x_1,\ldots,x_n])$ consisting of points which evaluate to "true" in an algebraically closed field over that point, and show by induction on the length of the formula that this subset is constructible.  The only tricky point is quantifiers, and these are handled by Chevalley's theorem.)
A: In line with Felipe Voloch's remark "Spec $\mathbb{Z}$ are where schemes really shine", I thought I'd also add this beautiful (expository and very readable!) paper of Serre:
"How to use finite fields for problems concerning infinite fields"
which makes crucial use of being able to do algebraic geometry over (finitely generated algebras over) $\mathbb{Z}$ in order to prove geometric statements (many of which are easily understable without any background in algebraic geometry!) over fields like $\mathbb{C}$ and $\mathbb{Q}$.
A: This is merely too long to be a comment.
I'm not sure that I completely agree with or understand the basis of the question.  Is this one of those questions where the OP is taking for granted that varieties are interesting and then wondering what schemes are useful for in this context?  If so, there have certainly been other questions of this sort on mathoverflow.  However, I get the feeling this is not what's being asked for, in which case I'm not sure that the example of character theory being used to solve a problem of group theory is completely analogous.  In that example, groups are already inherently part of character theory, so it seems reasonable that problems of group theory might be attacked using character theory.  When asking a similar question about schemes, I'm not sure what the starting point is.  To me, asking what you can do with schemes is more like asking 'what can you do with groups?' than 'what can you do with characters?'.  At this point, I would say that both groups and schemes are naturally occurring basic mathematical objects that are of great interest to many mathematicians.
A: Another answer to your question is provided by Mumford's book "Lectures on curves on an algebraic surface".   This book explains Grothendieck's proof (via schemes) of the fact that for a sufficiently positive curve $C$ on a smooth projective surface $S$, the algebraic equivalence class of $C$ modulo the linear equivalence class of $C$ has maximal possible dimension, namely the dimension of the Picard variety of $S$.
According to Mumford's introduction, the only known earlier proofs were analytic in nature.
Mumford's book is really excellent by the way; it is not a substitute for Hartshorne or any other introductory textbook, but is a wonderful "second course", which introduces ideas such as the Hilbert and Picard schemes, some deformation theoretic ideas, and related techniques, and (perhaps more than Hartshorne) really shows how you can use all of Grothendieck's new methods to actually do something!
Of course, this is not an elementary problem in the sense of your question, but in thinking about algebraic geometry and its foundations, its worth remembering that algebraic geometry already had an extremely rich history by the time Grothendieck introduced schemes, and so the
big outstanding problems (of which there were definitely many!) that demanded the introduction of this new technology were not simple ones (hence it's not so easy to justify the introduction of schemes with one or two very simple examples).  (If one wants a truly simple example, one can just discuss how the size of a fibre under a map of projective curves is constant, provided that one counts the size of the fibre using its scheme structure, i.e. taking into account the nilpotents that appear at ramified points.  But I don't know how compelling this example would be to non-algebraic geometers; it still may seem more like convenient book-keeping than a genuinely important new technique if you can't demonstrate it's utility through some specific application.)
A: Let me give two examples, however not so elementary,
1) The proof of the degeneration of the Hodge-De Rham spectral sequence by 
Deligne and Illusie in positive characteristic.
The basic assumption in Deligne-Illusie's theorem
is the existence of a lift over the Witt vectors $W_2$.
Hard to state within varieties. 
Moreover, Raynaud gives applications to instances of the Kodaira
vanishing in positive characteristic.
2) Grothendieck's study of the fundamental group.
The notion of an étale morphism can probably be discussed within varieties, but
one needs formal schemes, and the invariance of the fundamental group
by a nilpotent immersion.
But at the end, the result can be stated within varieties!
A: The "classical" example is surely duality of abelian varieties. If you want this duality to work over finite fields (or in characteristic p generally), it becomes apparent that you can't work with varieties alone. Technically taking the quotient by a group scheme that is not reduced is just too hard to express in the older geometric language. 
A: I should mention the Hamilton-Caylay theorem for matrices: proof: base change to an algebraic closure of your underlying field, and use the fact that diagonalizable matrices are Zariski dense. (However, this doesn't used scheme that are glued from affines.)
A: I am far from an expert in algebraic geometry, but I think that there is something that really should be said in answer to this question.
Namely, there are at least two big reasons one might introduce a new theory--one is to answer old questions, and the other is to ask new questions (often, of course, motivated by the old classics).  Of course, scheme theory was introduced to a large extent for the first reason.  But I, at least, think the greatest virtue of the theory is in the second area.
Almost every question we could have asked about complex varieties, or at best about varieties over perfect fields (or maybe dvrs, in the lead-up to the development of schemes), can now be asked about schemes over arbitrary rings (or non-affine bases).  We can see the geometry in diophantine equations; we can use cohomological tools to answer arithmetic questions.  We can analyze deep algebraic structures of commutative rings in a geometric way.  See e.g. Minhyong Kim's answer here.
I think this is what Kevin Buzzard's comment is getting at, in an extremely pithy way, in his comment about Fermat's last theorem; one might make a similar comment about, say, Falting's theorem.  In order to even ask the right questions to approach these classics, we must have access to geometry over all sorts of supposedly pathological bases:  imperfect fields, or even non-Noetherian rings.  100 years ago, who could have even imagined the statements of theorems about modular curves, for example, which seem totally natural today?
So back to your question:  what elementary questions can be addressed using scheme theory?  I guess I would say:  any question about families, all of arithmetic geometry, any question about varieties over $\mathbb{C}$ you might be interested in over another base, any application of cohomological methods from the analytic theory (e.g. Riemann-Roch) you want to generalize, almost any problem where moduli spaces come up, etc.
A: If $C$ is a sufficiently general rational curve in $\mathbb P^3$ of degree $d$, then the vector space of degree $k$ homogeneous polynomials vanishing on $C$ has dimension precisely equal to
$$\max\left\{0,\binom{k+3}{3} - dk - 1\right\}\;.$$
More geometrically, there exists a degree $k$ surface containing $C$ if and only if $\binom{k+3}{3} - dk - 1 > 0$.
This is a statement that could have been understood a hundred years ago. However, the proof involves degenerating rational curves to non-reduced schemes whose Hilbert functions can be computed more easily; see Hirschowitz's 1980 paper "Sur la postulation generique des courbes rationales."
A: This is not a great answer, but it was getting a bit long to be a comment, so I just made it community wiki instead.  If anyone (with rep >= 100) wants to elaborate, feel free to do so in the answer itself rather than in the comments.
There is a technique for showing a closed subset $Z$ of an (irreducible) variety $X$ is all of $X$ that does not seem to have any analogue without using schemes. Let's suppose that $Z \subset X$ has a natural structure as a closed subscheme that comes from its definition (the induced reduced structure won't work here, as will become obvious). To show $Z = X$, it suffices to show that $Z$ contains a nonempty open set.  If $z \in Z$, then to show $Z$ contains a neighborhood of $z$, it suffices to show that $Z$ contains every subscheme of $X$ supported on $z$--i.e., in a sense, $Z$ contains every infinitesimal neighborhood of $z$. A version of this is used in Mumford's book Abelian varieties to prove the Theorem of the Cube (II) in chapter III.
A: A smooth projective variety over $\mathbb{Q}$ has only finitely many places of bad reduction. Shimura had a horribly complicated proof of this in the language of Weil's foundations in a paper from the 50's. With schemes, it's completely obvious, as smoothness is an open condition. Even stating this without schemes is painful. The whole field of arithmetic geometry is an example of what you want. Actually, this is my only serious complaint about Hartshorne. He doesn't do any Number Theory and $\operatorname{Spec}\mathbb{Z}$ is where schemes really shine.
A: Purity theorem: A map between smooth complex algebraic manifolds of the same dimension has ramification locus of pure codimension 1.
One can prove this by a clever induction on dimension using punctured spectra of local rings and exact sequences in local cohomology both of which are difficult to deal with without schemes.
