Why Ramanujan graphs are named after Ramanujan?
To add some content to the answers here, I'll explain briefly what Ramanujan's conjecture is.

First of all, Ramanujan's conjecture is actually a theorem, proved by Eichler and Igusa.  Here is one way to state it.  Let $r_m(n)$ denote the number of integral solutions to the quadratic equation $x_1^2 + m^2 x_2^2 + m^2 x_3^2 + m^2 x_4^2 = n$.  If $m=1$, that $r_m(n) > 0$ was of course proved by Legendre, but Jacobi gave the exact count: $r_1(n) = 8 \sum_{d \mid n, 4 \not \mid d} d$.  Nothing similarly exact is known for larger $m$ but Ramanujan conjectured the bound: $r_m(n) = c_m \sum_{d \mid n} d + O(n^{1/2 + \epsilon})$ for every $\epsilon > 0$, where $c_m$ is a constant dependent only on $m$.  

Lubtozky, Phillips and Sarnak constructed their expanders based on this result.  I'm not familiar with the details of their analysis but the basic idea, I believe, is to construct a Cayley graph of $PSL(2,Z_q)$ for a prime $q$ that $1 \bmod 4$, using generators determined by every sum-of-four-squares decomposition of $p$, where $p$ is a quadratic residue modulo $q$.  Then, they relate the eigenvalues of this Cayley graph to $r_{2q}(p^k)$ for integer powers $k$.  

A reference, other than the Lubotzky-Phillips-Sarnak paper itself, is Noga Alon's brief description in Tools from Higher Algebra.