An action of a group $G$ on a set $X$ is a function $\alpha :G\times X\to X$ (colon-to notation), which is often written $(g,x)\mapsto gx$ (mapsto notation), with $\alpha$ omitted from the notation, such that

1. $ex=x$ for all $x\in X$, where $e$ is the identity, and
2. $g\left(hx\right)=\left(gh\right)x$ for all $g,h\in G,x\in X$, where $gh$ implicitly refers to the group operation in $G$ (also omitted from the notation).

Equivalently, via Currying, an action of $G$ on $X$ is a group homomorphism $G\to \text{Aut}\left(X\right)$, where $\text{Aut}\left(X\right)$ is the automorphism group of $X$ (so for sets, the group of all bijections $X\to X$, but phrasing the definition this way makes it natural to generalize to other categories). It's a good exercise to verify this; Arbital has a proof.

Group actions are used to make precise the notion of "symmetry" in mathematics.

Examples

Let $X=R^2$ be the Euclidean plane. There's a group acting on $R^2$ called the Euclidean group $ISO\left(2\right)$ which consists of all functions $f:R^2\to R^2$ preserving distances between two points (or equivalently all isometries). Its elements include translations, rotations about various points, and reflections about various lines.