Q:

How fast is the Bitcoin (mining) network? 

How fast is the Bitcoin mining network compared to other distributed computing networks? ;) 

A:

According to Bitcoin Charts, it is equivalent to 127.25 PetaFLOPS currently.  This would make it faster than 500 fastest supercomputers in the world (58.9 PetaFLOPS), or Folding@home (4.1 petaFLOPS) [reference].  However, this is only equivalent computation speed, if you would measure actual amount of FLOPS of Bitcoin mining it would be... 0, as Bitcoin does not use floating point operations.  Moreover, I'm not exactly sure how Bitcoin Charts computed that figure. 
However, objective measure of how fast Bitcoin is in computational sense, would be it's current hashrate - 9.738 Thash/s. At peak rate it was estimated to be over 15Thash/s.
As a side note, one might also be interested in the speed of propagation of blocks in Bitcoin network, which is estimated to be close to 2 seconds.