What is the most practical fully homomorphic cryptosystem? Craig Gentry recently gave the first fully homomorphic cryptosystem. Quite a bit of work has been done since extending his work. It seems, however, that no system is practical for real world use.


What are the current roadblocks making FHE impractical?
Which proposed system would you say is currently the most practical?
Bottom line. The short answer is that none of them are practical ... yet.  But there is a lot of active research, and if we're lucky, maybe that will lead to enough improvements that it might become practical.  We'll see.

More information. You can find some useful information at the following questions:


What Partial Homomorphic Encryption implementations exist and how do I leverage them? discusses some schemes that fall short of being fully homomorphic but are more practical and are enough to be useful in some settings.
In what ways does Full or Partial Homomorphic Encryption benefit the cloud? discusses applications of fully homormophic encryption and links to some papers that discuss the state-of-the-art and how practical the best schemes are.


These resources may be of interest:


Shai Halevi's Homomorphic Encryption Tutorial, presented at CRYPTO 2011.
Helger Lipmaa's collection of research papers on fully homomorphic encryption.


See also the following research papers:


Kristin Lauter and Michael Naehrig and Vinod Vaikuntanathan.  Can Homomorphic Encryption be Practical?
Jean-Sebastien Coron and David Naccache and Mehdi Tibouchi.  Optimization of Fully Homomorphic Encryption.
Marten van Dijk and Craig Gentry and Shai Halevi and Vinod Vaikuntanathan.  Fully Homomorphic Encryption over the Integers.
Damien Stehle and Ron Steinfeld.  Faster Fully Homomorphic Encryption.


This is a rapidly-evolving space, so anything I write here may become out of date quickly.  You will need to follow the research literature to learn the latest and greatest.