High-bandwidth Digital Content Protection (HDCP) is a digital copy protection system that is commonly associated with DisplayPort, DVI or HDMI connections.
Elgato Game Capture HD cannot receive any HDMI signal that has HDCP protection. It can receive HDMI signals that are not protected, like from the Xbox 360.
If you connect a HDMI signal that has HDCP protection, then Elgato Game Capture HD will not display video.
Elgato Game Capture HD can receive normal Component video signals, since HDCP is not used for analog video connections like Component.
Some products may use HDCP when you don't realize it. For example, many Android devices may use HDCP often.
iOS devices only tend to use HDCP when you are watching video you have paid for in some way (like from iTunes or Netflix). iOS devices don't use HDCP during gameplay.
More info about HDCP can be found here.
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