I sent one of my unicorns to explore and its icon doesn't move. The fog of war goes away, and I did encounter a beast on the island to battle, but the portal icon remains active (I can leave the island). It's very hard to explore the island when you don't know where you are.

I did explorations yesterday and they worked as expected (same computer - OS 10.10, Google Chrome).

Update - I went back to the game after putting my computer to sleep for ~30 minutes for my commute and the icon did move around when I tried the arrows again. Nothing changed (other than my internet connection being a different network), so I'm not sure how this fixed the problem, but figured I'd report it in case it shed any light.

Sorry to hear that you experienced this nasty problem. Please use hard refresh (usually Ctrl+F5) to get rid of the problem in case if you encounter it again. This will move you pet to the actual position and make it move as usual. It happens when something interferes the page load (bad server response on our side or bad internet connection on player's side) and some of the scripts aren't loaded fully. Unfortunately, it's not something that we can fix because it doesn't depend on us.

My dragons icon won't move

This happens to me often. As Angel said, simply refresh the page. I notice when my internet is being slow and the page hasn't loaded fully and I already hit buttons. Wait for the page to load fully before you begin, and refresh if it happens. This happens to me on all my browsers.