I have tried almost every stuff I searched on google. All of them were about using flash but apparently none worked. Can you suggest any method to do that? What is the one which bit.ly uses?
As far as I know, there's no other way right now to do it other than Flash because Firefox has better security when it comes to clipboard handling. Just look for updated scripts, as most of the flash scripts don't work with Flash 10+ (Flash 9 below only).
http://code.google.com/p/zeroclipboard/ - flash based, works with flash 10 (so it's bound to an element / CTA)
Here is an updated ZeroClipboard work with latest Flash 11. Javascript: Easy Cross Browser Copy to Clipboard Solution