Now for a third mathematical definition of irreducible, this time from Representation Theory:
    Definition Suppose (ρ,V) is a representation of some group G. If there is no proper non-trivial subspace W of V such that ρ(G)W is contained in W, then the representation is said to be irreducible.
Come to think of it, Representation Theory is repsonsible for a great many overloaded definitions; simple is also an equivalent condition to irreducible, then there's complete, regular and characteristic which spring instantly to mind which also have definitions elsewhere in mathematics.
What is nice about the idea of irreducibility is that for finite and indeed compact groups, every representation of the group can be split up into a direct sum of irreducible ones, where the number of such irreducible representations is equal to the number of conjugacy classes of the group.