(Very abstract algebra, category theory):
An exact sequence is a sequence of objects (which need to come from a special kind of category, a cartesian category; just ignore that and take them to be groups or rings or modules or something else you can imagine) and homomorphisms between them:

1=A₀ → A₁ → A₂ → ... → A_n → 1

1=A₀ → A₁ → A₂ → ... → A_n → ...

By abuse of notation, we write "1" for the trivial object (the group consisting of just the identity element, the ring {0,1}, etc.).

For the sequence to be exact, we additionally require of the homomorphisms that Im(A_n-1->A_n) = Ker(A_n->A_n+1).

This has immediate repercussions; see short exact sequence for the most important example.

Homology and cohomology "operators" (really sequences of functors!) transform short exact sequences into infinite exact sequences; this is commonly known as the snake lemma.