i had installed os lion recently after that my mac is not able to login with the apple id to activate imessage, face chat, gamecentre, app centre etc etc it is showing the following error "The server encountered an error processing registration."

HI,

There is this Apple document to work through:-

FaceTime, Game Center, Messages: Troubleshooting sign in issues

In addition to these things your Serial Number could be missing if the logic board has been replaced in a Repair or Refurbishment.

To Check go to the Apple Icon menu top Left and select "About This Mac"

In the panel that appears click twice on the line about the OS X version (It will change to the Build Number and then the Serial Number)

Since OS X 10.8.2 the "registration" process seems to require the Serial Number to be present.

It may be a way by Apple to check the software is running on a Mac.

7:55 PM Sunday; June 2, 2013

 iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.3)
 G4/1GhzDual MDD (Leopard 10.5.8)
 MacBookPro 2Gb (Snow Leopard 10.6.8)
 Mac OS X (10.6.8),
 Couple of iPhones and an iPad
"Limit the Logs to the Bits above Binary Images."  No, Seriously