orbit
Pick a point and let the whole group push it around. The set of every place it can land is its orbit — the “reachable region” of that point. If you imagine the group as a set of moves and the point as a token on a board, the orbit is exactly the collection of squares the token can ever occupy.
Formally, for an action of G on X and a point x in X, the orbit of x is G·x = { g·x : g in G }. Orbits partition X: two points lie in the same orbit precisely when some group element carries one to the other, and this is an equivalence relation, so distinct orbits are disjoint and their union is all of X. An action with a single orbit (every point reachable from every other) is called transitive.
Orbits are the natural “pieces” into which an action breaks a set. Counting them, or counting points within each, is the heart of combinatorial applications such as Burnside's lemma for counting colorings up to symmetry. The size of an orbit is governed by the orbit-stabilizer theorem: it equals the index of the stabilizer of any of its points, so for finite groups every orbit length divides the order of the group.
Let Z/4Z act on the four corners of a square by rotation. There is a single orbit of size 4: starting from any corner, the four rotations reach all four corners. If instead Z/2Z acts on the square's four corners by the half-turn, the corners split into two orbits of size 2 (each pair of opposite corners).
Rotational orbits of the corners of a square.