How can I coil up a rope so that it doesn't get tangled?
Around the elbow no good, for ropes they get tangled and for cords they get ruined over time.

See my instructions below, but check out this animation for reference.

Keep in mind, I am right handed and this is how I do it: 

Start by placing one end of the rope between your left thumb and forefinger, and grab the rope with your right hand about 2 1/2 - 3 feet away both palms down. While holding the rope in your right hand, grab it between your thumb and forefinger.  Now bring your right hand to your left hand so the tips of your thumbs meet (and the newly created loop should form towards your left wrist). At the same time, roll the cord that is between your thumb and forefinger on your right hand away from you.  Now grab it with your left hand and slide your right hand down the cord 2 1/2 - 3 feet and repeat till done. The last 2 or 3 feet will be used to wrap the cord/rope.

If done correctly, you'll be able to take the end you wrapped around the cord like in your pic, unwrap it, hold one end and throw the cord/rope to unroll.