Quick description
If you have an algebraic problem about the complex numbers, it might be possible to solve it by solving a similar, but easier, problem about finite fields.
Prerequisites
Basic notions of commutative and linear algebra (more sophisticated ideas are required to understand the proofs of the results quoted, but this is not needed to apply the trick).
This trick depends on the following proposition.
In fact for the first example below we only require the fact that has some maximal ideal such that the quotient is a finite field. The proof of this (and a fortiori of the first statement in the proposition) is not particularly easy to find in the literature. Some links are given below, and perhaps the standard reference would be Bourbaki's Algebre Commutative, Ch. V, Sec 3, no. 4, Corollaire I, p.64.
Example 1: AxGrothendieck theorem] The discussion here is cribbed from this article by Serre and a [http://terrytao.wordpress.com/2009/03/07/infinitefieldsfinitefieldsandtheaxgrothendiecktheorem/ blog by Tao
that followed from it, together with comments on that blog.
The proof is by contradiction. If is injective but not surjective then

vanishes whenever does and

there is some such that vanishes whenever does (i.e. never).
The rather odd way of expressing these statements is so that we may apply Hilbert's Nullstellensatz. The two statements imply, in turn, that

some power lies in the ideal generated by and

some power lies in the ideal generated by .
More concretely,

There is a polynomial such that \tilde Q(x) such that R\CP,Q\tilde Q\mathfrak{m}R\pi : R \rightarrow R/\mathfrak{m}. By Proposition 1 the image is a finite field \mathbb{F}\pi\mathbb{F}P\mathbb{F}^n\mathbb{F}^n, is injective but not surjective. This is manifestly nonsense on cardinality grounds.
General discussion
There are connections between ideas of this kind and logic/model theory, but this author is not qualified to discuss them. More details may be found in the "Princeton Companion" article by David Marker, where explicit mention of the AxGrothendick theorem is made on p642.