By extension, one speaks of the order of an element$x \in G$, as the order of the cyclicsubgroup$\langle x\rangle$ generated by the element. For example, the order of a permutation$\pi \in S_n$ is the least integer $1 \le k\le n$ such that $\pi^k = id$.

Sometimes one thinks of an infinite group as having order zero. The orders then have the natural order relation of divisibility?.

Other meanings

The term ‘order’ can also be used fairly generically as a synonym of ‘degree’ or ‘rank’, as in first-order logic, the order of a differential equation, etc. Of course, these various orders form a well-order, so this is not entirely unrelated either.