This answer no longer works as YouTube has deprecated the showinfo parameter.
You can hide the embedded player’s title bar by adding &showinfo=0. You cannot completely remove all the links to the original video. Here is the best you can do
<iframe width="560" height="315" src="https://www.youtube.com/embed/videoid?modestbranding=1&autohide=1&showinfo=0&controls=0" frameborder="0" allowfullscreen></iframe>
This code will remove the title bar, YouTube branding in controls, controls (optional, delete controls=0 if you need controls). But a white YouTube logo will be displayed on the video with the video link.
Update 1: Here is a new tool that I built to generate customized youtube embed player code- Advanced Youtube Embed Code Generator