Both the PlayStation 4 (PS4) and Xbox One can record gameplay via HDMI.
HDMI output from Next Generation consoles
Next Generation consoles have HDMI outputs.
When their output is encrypted with HDCP (High-bandwidth Digital Content Protection), then Elgato Game Capture HD will not be able to capture that content via HDMI.
When their output is not encrypted, then Elgato Game Capture HD will be able to capture content via HDMI.
Both consoles use HDCP protection when watching a purchased movie, or using services like Netflix or Hulu. Therefore, Elgato Game Capture HD won't be able to capture that content.
Xbox One: HDMI gameplay output is supported
The Xbox One does not have HDCP protection during gameplay, so it works fine with Elgato Game Capture HD.
PlayStation 4: HDMI gameplay output is now supported
The PlayStation 4 was released on November 15, 2013.
On day one, the HDMI output was encrypted using HDCP. That meant that video capture devices weren't able to receive gameplay video from it.
As of April 30, 2014, Sony released the 1.70 firmware update that allows users to turn off HDCP during gameplay.
Once you install this update, then connect your PlayStation 4 directly to your TV set or display via HDMI, without using Elgato Game Capture HD.
Visit the Settings > System section of the PlayStation 4 interface, and turn off Enable HDCP.
After that, you can use your PlayStation 4 with Elgato Game Capture HD.
Connect your PlayStation 4 to the HDMI In of Elgato Game Capture HD via an HDMI cable
Connect the HDMI Out of Elgato Game Capture HD to your TV or display via an HDMI cable.
Connect the Elgato Game Capture HD to your computer via a USB cable.
Component output from Next Generation consoles
Next Generation consoles do not have Component video output.