I believe only IE lets you access the clipboard. Older versions of other browsers used to, but this has been switched off by default as a security measure. Users can explicitly turn it on via settings/options/preferences, but this is obviously not ideal in most situations.
The workaround is to use a flash object on the page. Since Flash 10 added more security layers, user interaction is also required now with the flash object (e.g. a click rather than say onload event).
I found and implemented the good work from the well written article at the bottom of my answer. He explains the issue in more detail, with links to official statements from Adobe/Mozilla and supplies a usable and a downloadable example, and the source code to the fla. This is handy if you want to reskin/redesign his button.
I have tested successfully on Windows7 using latest (as at 7/7/2011) Chrome/Safari/RockMelt/FF/IE7/IE8/IE9 and MacOSX(SL) Safari/FF.
The only downside is that it uses flash which is mostly fine except for some mobile platforms and a small portion of (ab)normal users. Also I found you need to access over http (a web server), opening and using the demo page via the file system (i.e. double clicking the html file in Explorer) won't work.
Thanks for sharing Rahul, awesome job.
http://www.rahulsingla.com/blog/2010/03/cross-browser-approach-to-copy-content-to-clipboard-with-javascript