Ramanujan 691 congruence I know how to prove this congruence in two ways (one using basics of modular forms and the other using Hecke operators) and I will be working on proving other such congruences soon. 
The congruence states that for $n\geq 1$: 
$\tau(n) \equiv \sigma_{11}(n)$ mod $691$.
My main question is why this congruence is so important? I recognise it as a beautiful thing but the only reason I can come up with for it being interesting is that it links a geometrical function with a number theoretical function.
Are there any other reasons why this is interesting/useful?
 A: Congruences between Hecke eigenforms are outward, "physical" manifestations of corresponding relationships between the associated two-dimensional Galois representations.  
In this particular case, Ramanujan's congruence is related to the fact that
$691$ is an irregular prime (in the sense of Kummer).  Roughly the idea is that
Eisenstein series relate to reducible two-dimensional Galois representations, and cuspforms to irreducible two-dimensional Galois representations.  The existence of a congruence between the two points to the existence of an object
that is somewhere between reducible and irreducible: a certain reducible two-dimensional Galois representation which is, however, indecomposable.  The existence of this particular reducible, but indecomposable, two-dimensional representation shows that $691$ is an irregular prime.  
To see a hint of
how this could be, note that $691$ being an irregular prime means, by class field theory for $\mathbb Q(\zeta_{691})$ --- especially, the theory of the Hilbert Class Field --- that there exists an unramified abelian extension of $\mathbb Q(\zeta_{691})$; so irregularity of $691$ is related to the existence of a certain abelian extension of an abelian extension of $\mathbb Q$, and the
reducible but indecomposable Galois representation will have such a thing as its splitting field.
For more information on this, and related ideas, you might like to read Mazur's article on the subject; see the entry June 17, 2010: How can we construct abelian extensions on his web-page.
A: The congruence
$$
\tau(n) \equiv \sigma_{11}(n) \pmod{691}
$$
is one of several congruences that have been proven.
Now it is well known that
$$
\Delta(z) = 2 \pi^2  \sum_{n = 1}^\infty \tau(n)q^n.
$$
Lehmer conjectured that $\tau(n) \neq 0$ for all positive integers $n$. As far as I know this has been checked for $n < 10^{11}$. The congruences play a role in showing part of his conjecture.
As to why is it interesting to consider why $\tau(n) \neq 0$ for all positive integers $n$, see this.
A: This is an extract to Manin's paper "Periods of parabolic forms and p-adic Hecke Series":
"This congruence is so far, our only clue to understanding the 11-dimensional étale cohomology of the so-called Sato variety"
Even though I don't know what he means, he gives some references to Serre's articles.
