A "dovetail" drawer refers to the specific construction of the drawer; a dovetail joint is used to securely join the drawer side and the drawer front together. Dovetail drawers are secured together by a series of pins cut to extend from the end of one board interlock with a series of tails cut into the end of the other board. The pins and tails have a trapezoidal shape. Once glued, a wooden dovetail joint requires no mechanical fasteners (it's easier to see on the video than to explain)...
Dovetail drawers are easily recognized by their distinct dovetail joint at the outside corner of the drawer. These look like jigsaw puzzle pieces joined together at a 90 degree angle.
Kitchen cabinets that have drawers secured with nails can come apart with heavy usage. Dovetail joints are used to join the sides of a drawer to the front and back of the kitchen drawer, keeping the kitchen cabinet drawers strong and sturdy. A well-made dovetail produces a joint, that even without glue, can be difficult to separate.