Iwasawa's starting observation was that there are towers of fields in algebraic number theory, having Galois group isomorphic with the additive group of p-adic integers. That group, usually written Γ in the theory and with multiplicative notation, can be found as a subgroup of Galois groups of infinite field extensions (which are by their nature pro-finite groups). The group Γ itself is the inverse limit of the additive groups of the Z/pn.Z, where p is the fixed prime number and n = 1,2, ... . We can express this by Pontryagin duality in another way: Γ is dual to the discrete group of all p-power roots of unity in the complex numbers.
A first and important example is in terms of the field K = Q(ζ) with ζ a primitive p-th root of unity. If Kn is the field generated by a primitive pn+1-th root of unity, then the tower of fields Kn (inside C) has a union L. Then the Galois group of L over K is isomorphic with Γ, because the Galois group of Kn over K is Z/pn.Z.
In order to get an interesting Galois module here, Iwasawa took the ideal class group of Kn, and let In be its p-torsion part. There are norm mappings Im -> In when m > n, and so an inverse system. Letting I be the inverse limit, we can say that Γ acts on I: and ask for a description.
The motivation here was undoubtedly that the p-torsion in the ideal class group of K had already been identified by Kummer, as the main obstruction to the direct proof of Fermat's Last Theorem. Iwasawa's originality was to go 'off to infinity' in a novel direction.
In fact I is a module over the group ring Zp[Γ]. This is a well-behaved ring (regular and two-dimensional), meaning that it is quite possible to classify modules over it, in a way that is not too coarse.
From this beginning, in the 1950s, a substantial theory has been built up. A fundamental connection was noticed between the module theory, and the p-adic L-functions that were defined in the 1960s by Kubota and Leopoldt. The latter begin from the Bernoulli numbers, and use interpolation to define p-adic analogues of the Dirichlet L-functions. It became clear that the theory had prospects of moving ahead finally from Kummer's century-old results on regular primes.
The main conjecture of Iwasawa theory was formulated as an assertion that two methods of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was eventually proved in generality, for totally real number fields.