There are two ways to describe an object: how it is made and what it is doing. For example, a knife can be described as “an elongated flat piece of metal, sharpened on one edge, with a handle attached” (it is how it is made), or “a thing to cut bread” (how it is used).
I have not red every answer, but the discussion of groups in this thread so far appears to be restricted to the viewpoint of “how they are made”. But what do finite groups do?
They act. They act on sets of various nature; this sets are made of elements. The same group may have many different actions. For example, the group of symmetries of the cube can be seen as acting on
- the set of 8 vertices of the cube
- the set of 6 faces,
- the set of 8 edges seen as non-oriented segments
- the set of 16 oriented edges,
- the set of 4 non-oriented “main diagonals”,
- the set of 3 non-oriented axes passing through the centers of opposite faces.
Indeed, every symmetry of the cube is moving elements of each of this sets within that set (perhaps actually fixing some of them or even all of them).
Elements of the group, for the purpose of this discussion, can be called actors (I invented that name specifically for my post on Quora). What follows is a description, not a rigorous definition.
Each actor moves elements in the set in some way (and this could be an identity move, when nothing is actually changed in the set) . What follows are properties of actions of a group:
- There is an actor which does nothing.
- For every actor there is another actor, which reverses its moves.
- For any two actors, there is an actor who is doing the combination of movements of the first two actors.
An action of a group is called trivial if every actor does not move anything.
An action of a group is called faithful if different actors do different movements. In the example with the group of symmetries of the cube, actions 1 to 4 are faithful, 5 and 6 are not faithful.
The key point: if an action of a group is not faithful, the same movements of elements can be achieved by an action on the same set of another group with smaller number of actors.
Definition: A finite group is simple if all its actions are trivial or faithful.
In short, a simple finite group cannot be replaced, in its non-trivial action, by a smaller group.
This is why finite simple groups are atoms of finite group theory, and why classification of finite simple groups has tremendous importance for combinatorics.
As we can see, the group of symmetries of the cube is not simple. Moreover, its action 6 above contain only movements of 3 elements which can be done by the symmetric group Sym_3 on three letters, or, which is the same, by the group of symmetries of an equilateral triangle (this triangle can be easily seen within the cube). By contrast, the groups of rotations (symmetries which do not change orientation) of the equilateral triangle or the regular pentagon are simple.
By the way, “how it is made” and “what it is doing” are called, in Hegelian dialectics, essence and phenomenon. Questions “intuitively, what is …” refer to phenomena. Intuitively, a knife is a thing to put butter on bread. Intuitively, a group is a mathematical object that acts, or which can be used to describe action. There are other mathematical objects that also can act, in their own way: rings and algebras, for example. On the other hand, you can use a spoon to put butter on bread.