diagram chasing in homological algebra
salamander lemma $\Rightarrow$
four lemma $\Rightarrow$ five lemma
snake lemma $\Rightarrow$ connecting homomorphism
and
nonabelian homological algebra
Diagram chasing is a common technique in homological algebra for proving properties of and constructing morphisms in abelian categories, where one traces elements in various ways around commutative diagrams.
Many basic lemmas in homological algebra, such as the five lemma, the 3x3 lemma and the snake lemma, are typically proven by diagram chases. See for instance the proof at five lemma or any book on homological algebra.
The salamander lemma can sometimes be used to give more conceptual proofs.
There are at least five approaches to performing diagram chases in general abelian categories (not assumed to be concrete like Ab or $R$Mod), roughly listed in increasing order of sophistication:
Daniel Murfet, Diagram Chasing in Abelian Categories (2006) (pdf)
George Bergman, On diagram-chasing in double complexes (arXiv:1108.0958)