A highboy is a fancy chest of drawers that stands about as tall as a person. While the definition is a bit fuzzy, English has long recognized the 'tall thing with drawers' category. Many of us today would call a chest of drawers a 'dresser', but for a long time all of these bits of furniture were known as types of chest. An extra tall unit was called a 'high chest of drawers' or 'chest on stand' if it had long legs; or a 'double chest of drawers' or a 'chest-on-chest' if it had stubbier legs. The name highboy comes from a corruption of the French bois, meaning 'wood' and became common in English in the late 1600s to refer to an especially nice chest-on-chest, often with features such as a cabriole leg with a scroll foot or similar, and topped with cornices and finials.
As modern people have more choices in furniture styles and less interest in what was popular in France 400 years ago, the term has become more and more diluted, but the general tone of 'fancy and tall' remains.