Instructions
Test whether your Xbox 360, HDMI cable or TV is causing the problem. Try connecting the Xbox with a different video cable, test the HDMI cable with a different device if you have one, and try a different HDMI cable to see if it's the HDMI port on your TV that isn't working.
Make sure you connect your HDMI cable to an HDMI port. If your monitor only has a DVI port, the HDMI cable might not work, as without an HDMI port, the monitor will not be HDCP compliant. HDCP is a form of content-protection that requires HDMI.
Check your setup. It may seem obvious, but make sure your TV is displaying the right source, and that your HDMI cable is firmly inserted into both the back of your Xbox 360 Arcade and your TV. Unlike the VGA cable, the HDMI cable does not screw into place, and can come loose more easily. If your console displays flashing red lights around the power button, the HDMI cable is not properly inserted.
Reset the Xbox 360 display settings. This might be necessary if you last used your console with a different video cable or at a different resolution that your current TV does not support. Turn on your Xbox 360 console and hold down the right trigger button and the yellow "Y" button at the same time. Your Xbox 360 will detect the HDMI and reset the display output accordingly.
Return your Xbox 360 to the retailer where you bought it, or to Microsoft. If none of the above suggestions work, then you likely have a faulty Xbox and there is nothing that you can do to fix it without voiding your console's warranty.