A Perfecto is a figurado cigar with two tapered ends, narrowing at both the head and the foot while bulging in the middle. The shape is among the oldest in cigar manufacturing and the most demanding to roll well; an uneven taper or a poorly constructed bulge will produce a cigar that burns unevenly or draws inconsistently. A well-made Perfecto, by contrast, can be one of the most expressive cigars on offer.
The double taper changes how the cigar smokes. The narrowed foot makes the cigar light easily and smoke cool in the first third (less burning area means less heat). As the burn progresses into the bulge, the wider section produces more smoke and exposes more wrapper-to-filler ratio change, which often produces a complex middle third. The narrowed head then concentrates the smoke at the finish, giving the final third a focused intensity.
Notable Perfectos worth a try: the Arturo Fuente Hemingway Short Story (the canonical small Perfecto), the Padrón Family Reserve No. 50 Maduro, and the various Cuban Edición Limitada Perfectos when they appear.
For more on cigar shapes and construction, see The Anatomy of a Cigar.