Elgato HD60 S

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I've been toying with getting a HDMI video capture device for some time now.  Checking around I decided to go with an Elgato HD60 S - USB 3 capture device.  I considered the PCI-e options but I wanted the USB 3 option for my laptop use also.  I was also able to pickup the capture device locally so that was a nice bonus.  If you want a 4k capable capture card, the 4k PCIe device goes for around $400.  I got the HD60 S for $180 and if you want the PCIe HD60 Pro, it's normally +$20 for that PCIe model.

Now that you know about their capture line of products, we can get into usage.  There is a power connector USB type-C connector, HDMI input and a passthru HDMI output.  You can use the Game Capture HD software from Elgato to play your input device on your desktop monitor (as a window or full-screen).  I did notice the frame rate seems lower than if you play directly on your monitor by changing the display input to use the pass-thru (raw display).  Mostly I noticed this playing faster rate game likes Naruto Storm Trilogy.
This this being said, I did not notice any input lag or audio delay.  Just the seemingly lower frame rate, despite the device capturing @ 1080p 60 fps (frames per second).

If you like muxing audio with your game playing, I tend to open Game Capture HD or OBS Studio, so that I can play music while listening to in-game audio.  Basically what I did on the Xbox 360 as it had it's own media player.  If you would like to see video quality via streaming, check out my Twitch channel.
I started out streaming with the Game Capture HD software from Elgato and then switched to OBS Studio, once I figured out Sources control.

OBS Studio works very well with this.  I did notice a glitch where you want to have the HDMI device turned on, before opening OBS Studio or you will hear a weird static noise.  I would say OBS has better control options but the Game Capture HD is much easier to use.  A nice feature of the Game Capture HD Software is it will let you change your Twitch.tv info in the program, without needing to navigate your way to the Dashboard | Live settings on the twitch site, for each stream.
Speaking of Game Capture HD software, it will write temp files out while running, so be sure to have free space or change your settings to write them out to a disk with space, HDD or large SSD.  I'm writing to an HDD currently.

If you ever wanted to play something in VR on a big virtual screen like BigScreen or the Oculus Home environment, both of the viewer programs will go full-screen and you can play away with your headset on and potentially stream that environment out as well.

Game streaming is cool but since this is a HDMI capture device, you can also plug in a computer or laptop and capture that as well.  You can capture BIOS video as well but that will depend on your device and if it outputs POST info over HDMI.  I say this as my laptop does not do this, but my desktop will, so long as I don't have a DisplayPort monitor connected.  If you ever dealt with desktop capture software and found some of it wonky, this is a nice method to outright capture anything you would typically see on a display and record it to video or a stream.

If you want to blank the screen or show a title card, OBS Studio will allow you to do this while recording, if you wish to avoid doing it in post-editing.  I started dabbling with that on my twitch stream as well.

Wrapping up, this is a good device that works well.  I noted the framerate drop when playing on your desktop but want to note it's not unplayable at all.  Audio and Input button presses register without any lag I could notice.  The included software is quite nice and gets you started without having to learn the ropes around using OBS Studio for Streaming and Recording.
The USB HD60 S makes a great option if you ever wanted to be able to record a PC boot process and desktop environment without using desktop capture software, thus removing an OS-specific means to record.  Using the recording software and drivers, Elgato has native support for Windows and OS X.  I have not tried to record anything in Linux yet.

Using the output passthru port lets you play at native console resolution and frame rate, while you record or stream on your desktop.  It's a fun device and highly suggested if you want to get into some video capture for under $200 USD.
FYI this will not allow you to capture video from a HDCP device like a Google Chromecast.  You will see a black screen and the Elgato Game Capture HD and a red icon that says HDCP with a line going through a circle icon.  A Roku will allow you to record it at the OS-level but if you try to play video, it will honor the HDCP request not to show video and put an overlay over the video, noting HDCP lockout.  So long as HDCP is not restricting your device, you should have no issues capturing video.  Searching around it appears you have an option to disable HDCP for Playstation 4 consoles.  I do not have one so I cannot verify this but Elgato has a PS4 guide to disable HDCP.

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Bumping this topic as I still use Elgato Capture cards and enjoy them.

I have a desktop PCI 4K60 Pro https://www.elgato.com/en/game-capture-4k60-pro

In the desktop that works well, however I feel that the HD60 S+ and HD60 S (the HD60 S+ can capture up to 4k, where as S version is 1080P)

An extra bonus on the external USB3 devices besides being easily ready for use on a laptop as well, there is also a 3.5mm line input. You will need active HDMI video for the audio to be heard but this works great for me to stream vinyl records while I am recording or streaming.

HDCP protections are still as valid as they were with a PS4. HDMI splitters may help side-step that if you wish to force full streaming of a device, as I have done for my quest streaming to a local chromecast.