A box-pressed cigar has a flattened cross-section rather than the round profile most cigars carry. The shape is achieved by packing the cigars tightly into their box during the post-roll aging period and pressing them flat against each other; the cigars take on a square or rectangular silhouette that they retain after they leave the box. The technique is sometimes called the Cuban press or the Nicaraguan press, depending on origin and method.
The flat shape changes the smoke in a few subtle ways. Box-pressed cigars sit more steadily in your fingers and on a desk; they roll less. The burn often runs slightly slower because the filler is more compactly packed. Many smokers report that box-pressed cigars draw a little tighter than round cigars from the same blend, though this varies with rolling.
Padrón built its reputation on box-pressing, and the 1964 Anniversary Series and Family Reserve are perhaps the canonical examples of the style. Other brands that work in box-press include Oliva (the Serie V Melanio Maduro), My Father (Le Bijou 1922), and Illusione (the OneOff lines).
For more on cigar construction, see The Anatomy of a Cigar.