In his famous paper "The two cultures of Mathematics" T. Gowers gives examples of organizing principles in combinatorics.
(i) Obviously if events $E1, \cdots,E_n$ are independent and have non-zero probability, then
with non-zero probability they all happen at once. In fact, this can be usefully true
even if there is a very limited dependence. [EL,J]
(ii) All graphs are basically made out of a few random-like pieces, and we know how
those behave. [Sze]
(iii) If one is counting solutions, inside a given set, to a linear equation, then it
is enough, and usually easier, to estimate Fourier coefficients of the characteristic
function of the set.
(iv) Many of the properties associated with random graphs are equivalent, and can
therefore be taken as sensible definitions of pseudo-random graphs. [CGW,T]
(v) Sometimes, the set of all eventually zero sequences of zeros and ones is a good
model for separable Banach spaces, or at least allows one to generate interesting
hypotheses.
(vi) Concentration of Measure
More examples (by Tao and other) you can see at
http://ncatlab.org/davidcorfield/show/Two+Cultures
Do you know another examples in various areas? I mean, for example, globalization techniques in topology (structure functor in Hirsh, Differential Topology, $\S 2.11$ and Mayer–Vietoris sequence, in Bott & Tu, Differential Forms in Algebraic Topology $\S 5$).
So, many proofs look like "prove the local version of theorem and globalize".
Do you know such principles? It should be more specific than undergraduate course but it should be common used in your branch and be situated in "common wisdom" of mathematics.
Best Answer
The Choquet theory in convex analysis / functional analysis / whatever you want to call it. An element of a convex set should be some kind of "average" of extreme points. This has the status of a theorem for compact sets in normed linear spaces but is a useful guiding principle for not-necessarily-compact sets in not-necessarily-normed linear spaces. Chapter 14 in Lax's Functional Analysis book gives good examples of the wide array of applications of the same simple idea.