The Iwasawa main conjecture is a deep connection between two objects: class groups of cyclotomic fields and special values of the Reimann zeta function. The former object is algebraic: it is a class group of a number field. The latter is analytic: it is the value of a complex function at certain special points. The main conjecture, therefore (whatever it is), spells out an interesting connection between an algebraic and an analytic object.
The first indication that such a connection might exist is a theorem of Kummer.
Theorem. (Kummer) Let $p$ be a prime number. Then $p$ divides the class number of $\mathbf{Q}(\zeta_p)$ if and only if $p$ divides the numerator of $\zeta(r)$ for some integer $r=-1,-3,-5, \dots$.
As an example, set $p=691$. Then $691$ divides the numerator of $\zeta(-11)=\frac{691}{32760}$. So Kummer's theorem tells us that $691$ divides the class number of $\mathbb{Q}(\zeta_{691})$. This is already quite interesting because calculating class numbers by hand is basically impossible. This theorem says that the Reimann zeta function can be used to obtain this information in a different way.
But what is the meaning of $-11$ in $\zeta(-11)$? It turns out that $-11$ encodes the Galois action on the class group of $\mathbb{Q}(\zeta_{691})$. Precisely, the class group $\text{Cl }(\mathbb{Q}(\zeta_{691}))$ has a natural action of the Galois group $\text{Gal }(\mathbb{Q}(\zeta_{691})/\mathbb{Q})$. And this Galois group acts on $\text{Cl }\mathbb{Q}(\zeta_{691})$ via $\chi^{-11}$, where $\chi$ is the cyclotomic character. In other words, we have an inclusion of Galois modules
$\text{Cl }(\mathbb{Q}(\zeta_{691})) \supset (\mathbb{Z}/691\mathbb{Z})(-11),$
where the $(-11)$ means that Galois acts via $\chi^{-11}$.
This example actually holds in general, and this was proved by Herbrand and Ribet.
Theorem. (Herbrand-Ribet) Let $p$ be a prime number and let $r=-1,-3,-5, \dots$. Then
$\text{Cl }(\mathbb{Q}(\zeta_p)) \supset (\mathbb{Z}/p\mathbb{Z})(-r)\iff p \,\vert \, \zeta(r).$
The slogan here is:
$\text{zeta value}=\text{ class group with Galois action}.$
It actually turns out that much more is true. To see that, I'll rephrase the Herbrand-Ribet Theorem above as follows. Let $\text{Cl}^{(r)}(\mathbb{Q}(\zeta_p))$ be the $\chi^r$-eigenspace of $\text{Cl}(\mathbb{Q}(\zeta_p))$ (i.e: it is the subgroup of $\text{Cl}(\mathbb{Q}(\zeta_p))$ where Galois acts via $\chi^r$). Then Herbrand-Ribet equivalently says:
Theorem. Let $p$ be a prime number and let $r=-1,-3,-5, \dots$. Then
$\text{ord}_p(\#\text{Cl}^{(r)}(\mathbb{Q}(\zeta_p))) > 0 \iff \text{ord}_p(\zeta(-r))>0$.
In truth, however, it turns out that the $\text{ord}_p$ of both sides are in fact equal. That is, one can show that
$\text{ord}_p(\#\text{Cl}^{(r)}(\mathbb{Q}(\zeta_p))) = \text{ord}_p(\zeta(-r))$.
This however is much harder to prove, and requires the full strength of the Iwasawa main conjecture. In that sense, the Iwasawa main conjecture (whatever it is) is a strengthening of the Herbrand-Ribet Theorem.
To state the Iwasawa main conjecture, we have to do something which is a signature of Iwasawa theory: we have to state the formulate Herbrand-Ribet theorem in infinite towers. On the algebraic side, instead of looking at just one class group, look at class groups in an infinite tower of cyclotomic fields. On the analytic side, instead of looking at just one zeta value, we will look at infinitely many zeta values are $p$-adically interpolate them into a $p$-adic $L$-function.
For Herbrand-Ribet, we considered the group $\text{Cl}^{(r)}(\mathbb{Q}(\zeta_p))$, which is the $\chi^r$ eigenspace of the class group of $\mathbb{Q}(\zeta_p)$. Now we will do everything in $\mathbb{Z}_p$-extensions.
Consider the $\mathbb{Z}_p$-extension $\mathbb{Q}(\zeta_{p^{\infty}}/\mathbb{Q}(\zeta_{p})$ with layers given by:
$\mathbb{Q}(\zeta_p) \subset \mathbb{Q}(\zeta_{p^2}) \subset \mathbb{Q}(\zeta_{p^3}) \subset \dots \subset \mathbb{Q}(\zeta_{p^{\infty}})$.
Then for every $n \geq 0$, consider we can consider the $\chi^r$-eigenspace $\text{Cl}^{(r)}(\mathbb{Q}(\zeta_{p^n}))$. The central object of study will be the inverse limit of these groups:
$X_{\infty}^{(r)} = \varprojlim_n \text{Cl}^{(r)}(\mathbb{Q}(\zeta_{p^n})).$
Then $X_{\infty}^{(r)}$ is a module over $\Lambda = \mathbb{Z}_p \llbracket \text{Gal }(\mathbb{Q}(\zeta_{p^{\infty}}/\mathbb{Q}(\zeta_{p})) \rrbracket$. And $X_{\infty}^{(r)}$ is also finitely generated and torsion as a $\Lambda$-module. There is a non-canonical isomorphism $\Lambda \simeq \mathbb{Z}_p \llbracket T\rrbracket$. So by the structure theorem for finitely generated torsion $\Lambda$-modules, we have a pseudoisomorphism:
$X_{\infty}^{(r)} \sim \bigoplus_{i=1}^n \dfrac{\Lambda}{p^{e_i}} \, \oplus \bigoplus_{j=1}^m \dfrac{\Lambda}{f_j},$
where the $f_j \in \mathbb{Z}_p \llbracket T\rrbracket$ are distinguished polynomials^{1}.
Two quantities of interest:
Then we have Iwasawa's growth formula: there is an integer $\nu\geq 0$ such that
$|\text{Cl}^{(r)}(\mathbb{Q}(\zeta_{p^n}))| = p^{\mu p^n + \lambda n + \nu },$
for all $n \geq 0$. So the quantities $\mu$ and $\lambda$ in the structure theorem have a concrete meaning in terms of class numbers of the intermediate layers of the tower.
We can wrap up $\mu$ and $\lambda$ in a quantity called the characteristic ideal. The characteristic ideal of $X_{\infty}^{(r)}$ is defined as:
$\text{char }X_{\infty}^{(r)} = (p^{e_1 + \dots + e_n} f_1 \dots f_m) \subset \Z_p \llbracket T \rrbracket$.
That is, you take all the "denominators" in the structure theorem, multiply them together. The ideal generated by that element in $\Z_p \llbracket T \rrbracket$ is the characteristic ideal. Note that we can recover $\mu$ and $\lambda$ from the characterstic ideal.
On the algebraic side, we looked at infinitely many number fields and "strung together" their class groups by taking an inverse limit. On the analytic side we will look at infinitely many zeta values and we will string them together by $p$-adically interpolating them. Doing this will create a $p$-adic $L$-function, which is the main object of the analytic side of the main conjecture.
We discussed $p$-adic $L$-functions in detail in a previous post. I'd definitely recommend reading that post first before coming here. For here, I'll just recall the main theorem from that post, which proves the existence and uniqueness of the $p$-adic Riemann zeta function.
Theorem. (Kubota-Leopoldt) Let $p$ be a prime, and let $r = -1, -3, -5, \dots$ be a negative odd integer. There exists a unique $p$-adically continuous function $\zeta_p^{(r)}: \Z_p \to \Z_p$ with the following interpolation property:
Define the set $S_r = \{ n \in \Z_{<0}: n \equiv r \mod (p-1) \}$. Then for all $s \in S_r$, we have
$\zeta_p(s) = (1-p^{-s}) \zeta(s)$.
This function is called the ($r$-th branch of the) Kubota-Leopoldt $p$-adic zeta function. The set $S_r$ consists of the "points of interpolation" of $\zeta_p^{(r)}$. At these points, the function $\zeta_p^{(r)}$ coincides with the Riemann zeta function with the Euler factor at $p$ removed.
If any of this feels unfamiliar, check out this post where I explained this theorem much more slowly.
The Iwasawa main conjecture relates the module $X_{\infty}^{(r)}$ to the $p$-adic $L$-function $\zeta_p^{(r)}$:
$X_{\infty}^{(r)} \xleftrightarrow{\text{Iwasawa Main Conjecture}}\, \zeta_p^{(r)}$.
The problem is that these two objects live in different worlds: the module $X_{\infty}^{(r)}$ is an object from commutative algebra whereas $\zeta_p^{(r)}$ is a $p$-adic analytic function. So it's not even clear how to state a precise conjecture relating them. It was Iwasawa who realized that $\zeta_p^{(r)}$ can be expressed as a power series in $\mathbb{Z}_p \llbracket T\rrbracket$, which will allow us to view it as an object in commutative algebra.
Theorem. (Iwasawa) Let $p$ be a prime and let $r=-1, -3, -5, \dots$ be an odd negative integer. Then there is a unique power series $\zeta(T) \in \mathbb{Z}_p \llbracket T\rrbracket$ satisfying
$G_r((1+p)^s-1) = \zeta_p^{(r)}(s)$.
for all $s \in \Z_p$.
By abuse of notation, we also call $G_r(T)$ the ($r^{th}$ branch of) the Kubota Leopoldt $p$-adic zeta function. We will no longer think of the $p$-adic zeta function as a literal function $\Z_p \to \Z_p$, but we will view it as a power series in $\Z_p \llbracket T \rrbracket$.
On the algebraic side, we have the characteristic ideal $\text{char }(X_{\infty}^{(r)})$. On the analytic side, we have the $p$-adic zeta function $G_i(T)$. We can look at the ideal $(G_i(T)) \subset \mathbb{Z}_p \llbracket T\rrbracket$ generated by this $p$-adic zeta function. The main conjecture asserts that these two ideals are equal.
Iwasawa Main Conjecture. Let $p$ be a prime and let $r = -1, -3, -5, \dots$ be a negative odd integer. Then:
$\text{char }(X_{\infty}^{(r)}) = (G_r(T))$.
When I first saw this, I understood basically nothing. So I want to take it apart a bit to understand it better. First, I found it helpful to phrase this in terms of power series instead of ideals. We had a pseudo-isomorphism:
$X_{\infty}^{(r)} \sim \bigoplus_{i=1}^n \dfrac{\Lambda}{p^{e_i}} \, \oplus \bigoplus_{j=1}^m \dfrac{\Lambda}{f_j}.$
Also, by the Weierstrauss preparation theorem, we can factor $G_r(T)$ as
$G_r(T) = p^k \, u(T) \, f(T),$
where $u(T)$ is a unit and $f(T)$ is a distinguished polynomial.
Iwasawa Main Conjecture. (Power Series Formulation) We have
$p^{e_1 + \dots + e_n} f_1 \dots f_m \, \sim \, G_r(T)$.
where $\sim$ indicates equality upto a unit in $\Z_p \llbracket T \rrbracket$. In particular, $e_1 + \dots + e_n = k$ and $f_1 \dots f_m \sim f$.
This might seem kind of opaque (I definitely thought so at first), but it has a very concrete consequence about $\mu$- and $\lambda$-invariants. To see it, let's remember Iwasawa's growth formula: there is an integer $\nu\geq 0$ such that
$|\text{Cl}^{(r)}(\mathbb{Q}(\zeta_{p^n}))| = p^{\mu p^n + \lambda n + \nu },$
for all $n \geq 0$. The main conjecture then implies that:
Consequence of IMC.
So we can read off $\mu$ and $\lambda$ right from the $p$-adic zeta-function. This is really weird! The $p$-adic zeta-function interpolates special values of the Riemann zeta function. Why on earth should we be able to read off $\mu$ and $\lambda$-invariants for class groups from these zeta functions?
Regardless of how mysterious the statement is, Mazur and Wiles were actually able to prove the main conjecture in a brilliant 1984 Inventiones paper.
Theorem. (Mazur-Wiles) The Iwasawa Main Conjecture is true.
A polynomial $f \in \mathbb{Z}_p \llbracket T\rrbracket$ is distinguished if when you reduce it mod $p$, only the highest degree term remains. ↩