The verb worry is an anxious word; it means to be concerned or nervous. If you sent your carrier pigeon out in the morning to deliver a message, you might worry if it hadn't returned by the afternoon.
When you worry, you feel uneasy — the way you might worry that you've got spinach in your teeth when you're at a job interview. When it's an animal doing the worrying, it takes on a different meaning: a dog will worry a bone — or gnaw on it and play with it, for hours. The word worry comes from the Old English wyrgan, which originally meant "strangle," and changed over the years to mean first "harass," and then "cause anxiety to."