Cycle notation is a compact way to write permutations. Every permutation can be uniquely written as a product of disjoint cycles. Every permutation is a product of transpositions, and the parity (odd or even number of transpositions) is invariant.
From your work with permutation groups, you know that a permutation is a bijection from a set to itself — a relabeling of positions. Writing out a permutation as a full two-row table (top row: original positions, bottom row: where each goes) is cumbersome for large sets. Cycle notation compresses the information by describing orbits: chains of elements that get cyclically mapped into one another. Write (1 3 5) to mean 1 → 3 → 5 → 1. Every element not listed is fixed.
Two cycles are disjoint if they involve completely separate elements. The fundamental theorem of cycle decomposition says every permutation factors uniquely (up to ordering the cycles and starting position within each cycle) as a product of disjoint cycles. For example, the permutation sending 1→3, 2→2, 3→5, 4→1, 5→4 in S₅ decomposes as (1 3 5 4)(2). Disjoint cycles commute — you can apply them in any order because they act on different elements. This makes composition of disjoint cycles trivial: just chase each element through its own cycle.
A transposition is a 2-cycle (i j), swapping two elements and fixing all others. Any longer cycle decomposes into transpositions: (a₁ a₂ ... aₖ) = (a₁ a₂)(a₁ a₃)···(a₁ aₖ). This decomposition is not unique — there are many ways to write a given permutation as a product of transpositions. But here is the invariant: the parity of the number of transpositions is always the same for a given permutation. A permutation is even if it requires an even number of transpositions, odd if an odd number. A k-cycle is even when k is odd, and odd when k is even. (A 3-cycle is even because it takes 2 transpositions; a 2-cycle is odd because it takes 1.)
Parity is the key to the sign of a permutation, the determinant formula for signed permutations, and the definition of the alternating group Aₙ (the subgroup of all even permutations). The sign function sgn: Sₙ → {+1, −1} is a group homomorphism, with kernel Aₙ. The fact that parity is well-defined — that no permutation is simultaneously even and odd — requires proof, but the intuition is that every algebraic operation that "toggles" parity must do so consistently. This invariance will be the engine behind the sign of a permutation and eventually the Leibniz determinant formula.