This is my first major experience with a front loader, in fact! There’s one in my childhood home, but it’s relatively old and generally handles a larger household, so obviously this doesn’t come up as much.
And here I thought I was being clever just washing a few things. Thanks for the info!
There are two separate things: 1. If the washing machine is level. 2. If all four corners are firmly set on the floor: sometimes only three are on the floor and the fourth has a gap–so you can rock the machine a little in one direction.
I’ve never had a front-loading washer, but even with a top-loader:
ETA: But first, like the others are saying, make sure all four feet are firmly on the floor. They should all be adjustable.
Update: the washer still appears to be moving, regardless of how big the load is. As far as I can tell, the front feet are firmly on the floor. I can’t see the back feet, of course, but when I press down on the back it doesn’t seem to move.
How old is the washer?
Most washers have some type of damper that helps limit the amount of vibration that the spinning drum creates. If those are broken, even a nominally balanced load might cause the machine to walk.
Just over six years old, apparently. A Samsung VRT.
Some good comments (now, that’s a surprise…) here:
Yeah, definitely not a place I would have considered looking for good info. 
One mentioned that there needs to be at least an inch of space between appliances and walls. I don’t have that (between the washer and dryer); wonder if that could help? I’m not very strong, though, so I’m not sure about moving the washer for whatever purpose, like anti vibration pads. Hm.