Hi there, can anyone explain to me what the "Lefschetz Principle" is by some clear "classical" examples (not relying explicitly on model theory, say). Thanks !

Considering your choice of tags, it's plausible that you actually mean the Lefschetz principle. In its vaguest form, this principle says that algebraic geometry behaves "the same way" over $\mathbf C$, over an arbitrary algebraically closed field of characteristic zero, and over an arbitrary algebraically closed field of sufficiently large characteristic. This is typically used (e.g. in conjunction with GAGA) to deduce algebraic statements from things proved analytically over the complex numbers. See What does the Lefschetz principle (in algebraic geometry) mean exactly? for more information. On the other hand, you might also be referring to H. Davenport, "On a principle of Lipschitz", 1951. 


The simplest example of an application of the Lefschetz principle is to prove the residue theorem (in characteristic zero): The sum of the residues of a differential on a smooth projective curve is zero. If you are given a curve and a differential over a field of characteristic zero, there is a finitely generated field over $\mathbb{Q}$ where the curve, the differential, the poles and the residues there, are all defined, so take this field and embed it in $\mathbb{C}$. The residue theorem over $\mathbb{C}$ is, of course, an immediate consequence of Cauchy's theorem. You can even deduce the positive characteristic case from this with a bit more effort. 


HarishChandra uses this buzzword also as a guiding principle in the sense: Every thing what is true for reductive Lie groups, should be expected to be true for $p$adic reductive algebraic groups as well. I actually thought of asking a similar question a while back. I figured out later the same thing what Dan Petersen is mentioning, the terminology seems to originate from algebraic geometry. 

