Activated charcoal/carbon is basically just a structural carbon that is highly porous and sponge-like. It has a massive surface area for its volume and can trap other materials and act like a filter. It can remove smells and organic molecules that can make the water taste funny. You can easily make your own activated charcoal from a variety of materials and in pretty large quantities. There are plenty of videos on YT that show the process, but here's one from Cody's Lab (youtube.com) to get you started if you want to do this.

DE is the finest filtration you can achieve. Better than sand. It would need to be in a separate container with a way to transfer the water between barrels. DE works best in a grid system, which needs a pump to circulate the water between the grids.

Edit: water filtration DE is different that agricultural DE, and I’m not sure either should be sprinkled inside a chicken run if that’s what you’re thinking.