A bonnet is an old-fashioned hat that ties under the chin. Sometimes people dress babies up in frilly clothes and bonnets.
A bonnet isn't stiff like many hats are — instead, it's made of soft fabric and has no brim. It was common during the 17th and 18th centuries for women to wear bonnets, to keep their hair tidy and protected from dust and sun when they were outdoors. Today, you're less likely to see a bonnet on a grown woman, unless she's a historical reenactor or acting in a play. The root of bonnet is the Scottish bonat, "brimless hat."