What algorithm on a PRAM computes the connected components of a graph with least time complexity?
As Jeff suggested, I moved my comment to the answer section: 

On the CRCW PRAM, achieving  $O(\log n)$ time is fairly easy (see J. v. Leeuwen, Handbook of TCS, Chapter 17). 

On the EREW PRAM, Chong, Han and Lam (JACM, Vol 48(2), 2001) give an $O(\log n)$ algorithm for MST on an EREW PRAM. This should also solve the CC problem. Alternatively, use Reingold's logspace algorithm for undirected connectivity and standard simulation methods.