Motion Photo is a camera feature that Samsung introduced to its phones back in 2016 with the Galaxy S7 and Galaxy S7 edge. Similar to Apple's Live Photos, Motion Photo captures a few seconds of video before you tap the shutter button to take a photo. It gives you the ability to see and save that moment in moving form and also lets you to “pinpoint which moment you want to preserve as the still frame”, as Samsung puts it.
Initially, the Motion Photo feature was hidden in the camera settings on all Galaxy devices, but Samsung relocated the toggle for enabling and disabling the feature with One UI 2.0, leading some to think that it was removed. However, Motion Photo is still there: To enable it, you just have to hit its toggle in the camera viewfinder. It's the icon of a rectangle with a play button, right next to the button that lets you change the aspect ratio of your photos, as you can see in the screenshot below.
While the method of toggling Motion Photo on and off has changed, the feature itself works exactly the same as it has always done. You get up to 3 seconds of video leading up to the moment you tap the shutter button. It would be neat if Samsung would add the option to capture video before and after the shutter button is pressed, which is how Apple's implementation of moving photos works, but given how long Motion Photo has been around, it's unlikely Samsung has any plans to change things up.
Note: The Motion Photo feature is not available on all Galaxy devices running One UI 2.0 or One UI 2.1, only flagships and some upper mid-range models.