If you did a proper clone you will able option key boot to the external drive then use Disk Utility to clone the external to the internal following this video.
If you just opened Finder and drag and dropped Macintosh HD, you will not be able to make a bootable restore from that. That would copy over your personal data though, like documents and music etc.
You will need to reinstall the OS make a user account then you can copy your personal data over. What year and model Mac do you have?
If you already have the OS installed, say like Mavericks, yes you can command-r boot to the hidden recovery partition on the drive and reinstall the OS that way. But if there is no recovery partition on the drive you would need to use Internet recovery and you Mac does not support that.
There is a pretty good article here that explains it. The article is for Mountian Lion, but it still works the same way.