Let \(\varepsilon\colon S_n\to\theset{\pm1}\) be the sign homomorphism and \(\sigma\in S_n\). The permutation \(\sigma\) is even if \(\varepsilon(\sigma) = 1\) and odd if \(\varepsilon(\sigma) = -1\).
Since every permutation is a product of transpositions and each transposition has sign \(-1\), a product of \(t\) transpositions has sign \((-1)^t\). So \(\sigma\) is even if and only if it is a product of an even number of transpositions, and odd if and only if it is a product of an odd number of transpositions.
What distinguishes an even permutation from an odd one?