If you're concerned, you're feeling worried. When your clumsy friend announces his plans to become a stunt double, you should be concerned for his safety, and if you're not, you're a lousy friend.
The adjective concerned has roots in Latin terms for seeing clearly, as well as for being in relation to something. If you feel concerned, say for your neighbor who just bet his savings on a horse race, that's because you truly care for his well-being and you see that he's doing something stupid. Sometimes we use concerned to show the limits of our investment. As far as you are concerned, Madame X can jump off a cliff! You don't care!