(Mathematics, Group Theory:)
I like thinking about group actions as a sort of applied group theory — if there were Platonists about, I'd say it was when the group stops being a bum on its couch in the ideal world of maths and goes out into the world. Or, in another sense, its when a group starts acting, instead of just sitting there and looking pretty.
Groups act upon sets. A (left) group action is a function taking an element of the group acting and an element of the set being acted upon. The action (which we'll label as ⋅ : (<group>, <set>) → <set>) determines how the group interacts with the set, but to preserve the properties of the group, group actions have to satisfy two properties. If g and h are arbitrary elements of the group, and x is an arbitrary element of the set, then the two properties of the group action can be written:
- g⋅(h⋅x) = (g*h)⋅x
The action respects the group's operation.
- 1⋅x = x
Where 1 is the identity of G; this means that the action respects the group's identity.
From these one could also prove that the group action respects the inverses of the group. So a group action sort of transfers the group's structure onto a set. There are also right group actions, where the order of arguments are flipped. Right. Examples.
Say you have a set of letters, {a, b, c, ... z}. We can swap these letters around by having the symmetric group with 26 elements act upon the letters in the usual way by mapping a to 1, b to 2, and so on, and having the permutations act as such. This is basically what happens whenever you swap letters about while playing Scrabble, but it's done so unconsciously that considering it to be a "real" group action might be a bit of a stretch. But it's because of this overused action that we can count the number of ways to interchange and order a set with n unique elements — there are n! (n factorial) ways.
This particular class of group actions apparently has a lot of applications to chemistry — I think something to do with the various symmetries of the molecular structure of chemicals — and physics.
The linear transformations of euclidean spaces can all be represented in matrix form. The invertable square matrices of a particular dimension are called the general linear group, so the rotating or translating a euclidean space (something graphics programmers are intimitately familiar with) is essentially a group action from the general linear group onto whatever euclidean space is of interest.
Since every group has an underlying set, groups can act on themselves. One interesting way they can act on themselves is by conjugation: g ⋅ h = g*h*g-1. This type of action turns out to play a big role in group theory; it's used in working out the details of Sylow's theorem. Also, normal subgroups can be described as subgroups closed under conjugation.
Another way groups can act on themselves is by just the usual left-multiplication: g ⋅ h = g*h. Working out the details of this sort of group action proves that every group can be represented as a permutation group, a result also known as Cayley's Theorem.
Unfortunately, Cayley's theorem for groups isn't terribly useful, but it has analogs for other structures (like ordered groups and topological groups) that are relatively useful in group theory.
Group actions are terribly natural things, once you get used to them. The permutation example in particular should seem very natural: no matter what you're rearranging, the only thing that really changes the ways they can be rearranged is the number of unique elements available. For a field that has unjustly obtained a reputation for being obscure, abstract, and useless, practical application of group theory abound — mainly through group actions.
References: Dummit & Foote's Abstract Algebra, 3rd ed.