Let me try and add to Georges already great answer.

Let me, perhaps at risk of my own peril, try and summarize the very basic idea in one sentence:

Solution sets $S$ of equations ought to have intrinsic geometry which informs us of the nature of $S$.

Now, while this is nowadays a commonplace ideology, let me point out one tiny bit of subtlety in my above statement. Namely, solution sets of what type of polynomials, and solutions where? Classically one would interpret this sentence as being shorthand for something like:

Solution sets $S$ of sufficiently nice equations over the real (complex) numbers have intrinsic geometry which informs us about $S$.

Now, this is totally believable. If one interprets 'sufficiently nice' correctly then these solutions sets will be, say, real (complex) manifolds which, of course, have intrinsic structure, and yes, this structure tells us something about $S$.

That said, one of the basic tenets of number theory is that, sometimes, it's more interesting to consider solutions of equations not over the real or complex number but over objects with much richer arithmetic theory. Perhaps this means looking for solutions over a 'non-geometric' field like $\mathbb{Q}$ or $\mathbb{F}_q$, or perhaps, even over a ring like $\mathbb{Z}[i]$.

Secondly, in number theory we are often times interested in types of equations which are not sufficiently nice. This is a geometric term and, ostensibly, number theory is non-geometric, so such requirements seem unnatural.

Thus, reevaluating my first statement one then may begin to balk—such equations over such general rings have **no right** to have geometric structure. Where is such a geometric structure coming from? Certainly not from the underlying rings—the ring $\mathbb{Z}[i]$ doesn't carry sufficiently rich geometric structure to be able to talk about curve theory. What should it mean to consider the *cotangent bundle* of a set of solutions to an equation over $\mathbb{Z}[i]$? What should it mean to take its singular cohomology? What is a 'compact Lie group' over $\mathbb{Z}[i]$, and should it have a structure theory?

The realization of algebraic geometers (including Grothendieck) was that the equations **themselves** had intrinsic geometry. Then, any geometry on the solutions sets over some given ring are just derivative of the geometry of the underlying equations. This also does away with the fear of doing geometry over something ungeometric such as, say, $\mathbb{F}_q$, since it's the equations themselves providing the geometry.

Or, thought about in a more Grothendieck-ian (like Dickensian?) way the geometry comes from the collection of the solution sets of the polynomials over **all** rings, and how these sets vary with the solution ring. Said differently, if $X(R)$ denotes the solution set of the polynomials in $S$ over a ring $R$, it's not the set $X(R)$ that has a geometry but the functor $X$ itself that does.

Grothendieck and co.'s great innovation was realizing how to put this philosophy on firm footing. One needs to backup such a brash statement that the equation $x^n+y^n=z^n$ has intrinsic geometry, and a geometry sufficiently rich to be able to say something interesting about its set of solutions in some ring $R$. And, as Georges indicates it requires an extremely formidable amount of technical machinery to do this.

Now that one has this all out of the way all of your questions fall sort of neatly in line:

- By applying geometric intuition/tools, amongst the most powerful and easily intuited amongst all of a mathematicians toolbox, to something which (recycling my above phrase) ostensibly has no right to be amenable to such techniques (e.g. the equation $x^n+y^n=z^n$ over $\mathbb{Q}$).
- I think it's a little bit of black magic. One might go to bed every night dreaming of the intrinsic geometry of equations, but without deep technical understanding of the subject and a shifting of ideas (the Grothendieck/Yoneda type philosophy) this seems doomed. This is what makes it so beautiful and exciting.
- All of them. When talking to a fellow number theorist I can say the words fundamental group, cohomology (of a constant sheaf!), cotangent bundle, smooth, Lefschetz fixed point formula,... and I needn't specify if I am talking about an elliptic curve over $\mathbb{F}_p$ or a smooth surface over $\mathbb{C}$. The utility of this geometric language is not to be understimated. Even though it's a very, very simple instance, this recent answer of mine gives a great example of how geometric language makes purely number theoretic questions fall in line with there geometric counterparts.
- No, I don't think this is the case. But, without diving headfirst into the subject I don't think you can appreciate the fluidity and power that Grothendieck's uniform language brings to much of algebraic geometry, whether over $\mathbb{Z}$ or $\mathbb{C}$.