I have two coins I found in pocket change and both look to me to have errors on the reverse. The NH coin looks like something extra was there (you can see by the 2000 there is a shiny blotch). Mostly it's half of the motto is missing. The NJ coin is stamped pretty deeply which doesn't show well in the photo, but a lot of the coin didn't even stamp, like the state name, the motto and some of the main image. I'm not much of a collector, but are these worth getting appraised or encased?
Those are struck-thru grease or dirt errors, where dirt or grease gets stuck in the die. They usually don't command a premium, if any, so not really worth getting encased.
Yes, much of the missing elements is the result of a grease-filled die. Where you see the splotches to the left of and above "2000" and possibly to the right of "1788" looks like they may be the result of a strikethrough from debris falling from the die. Chris