The computers: Sage and Selmer
Main story: A Trillion Triangles
The computers used in the congruent numbers calculation are powerful general purpose computers, but they are far from what would be called a "supercomputer" today.
Selmer4 x Quad core AMD K10 @ 2.4 GHz
1.5TB Hard Drive
The precise configuration of Selmer is given on the manufacturer's website.
Note the Raid array was important. There are 4 drives of 0.5TB but one drive is used for parity, which means errors are detected.
The ECC memory is similarly important. It has active error correction and can correct bits flipped by cosmic rays (about 4 per day for a system this size).
Selmer is funded by Warwick University Department of Mathematics and by the Engineering and Physical Sciences Research Council in the UK, under William Hart's EPSRC grant "Algorithms in Number Theory".
SageActually, the Sage computer is now part of a cluster of computers, and two of them, mod and geom were used in this calculation. They have identical configurations:
4 x 6 core Intel(R) Xeon(R) CPU X7460 @ 2.66GHz
The precise configuration of mod and geom is given on the manufacturer's website.
Sage is funded by the National Science Foundation in the US, under grant DMS-0821725.