A sequence of Homomorphisms

$... \to A_{n+1} \to_{\alpha_{n+1}} A_n \to_{\alpha_n} A_{n-1} \to ...$

is said to be exact if $Ker{\:\alpha_n} = Im{\:\alpha_{n+1}}$ (where these refer to the Kernel and the image).

This chain is a Chain complex, whose Homology groups are trivial.

In Homology theory, exact sequences help in determining the relation between homologry groups of subspaces.

See here for some properties and definition of Short exact sequence

Homology long exact sequence

Theorem 2.13 (book) gives a long exact sequence for the Reduced homology groups of a space, subspace, and quotient space.

Long Exact Sequence of Pair

short exact sequence of chain complexes. when we pass to homology groups, this short exact sequence of chain complexes stretches out into a long exact sequence of homology groups

https://en.wikipedia.org/wiki/Exact_sequence