Chasing a ring is usually done with woods like osage, locust, mulberry, etc where you have a definite sapwood and heartwood. When removing the sapwood to get to the heartwood for your bow you begin removing growth rings, one at a time until you get a good clean heartwood ring that will be your bows back. The act of removing growth rings, one at a time, is chasing a ring.
You can chase a ring with any wood. All trees have yearly growth rings. With white wood, like maple, if you cut the wood during the growing season, you can peel the bark to reveal the ring that will be your bows back. With this method, you don't have to chase a ring. If the ring under the bark is damaged or otherwise unusable, you can chase the next best ring for the bows back.