A conical hat is a pointed hat.
They are most often worn by the magical dwarfs, witches and wizards. However, you are welcome to wear them as well.
Essentially, "conical" means having the shape of a cone. A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
Pointed hats have been a distinctive item of headgear of a wide range of cultures throughout history. Though often suggesting an ancient Indo-European tradition, they were also traditionally worn by women of Lapland, the Japanese, the Mi'kmaq people of Atlantic Canada, and the Huastecs of Veracruz and Aztec (illustrated e.g. in Codex Mendoza). The Kabiri of New Guinea have the diba, a pointed hat glued together.
The Asian conical hat, commonly known as an Asian rice hat, coolie hat (in the UK), or farmer's hat, is a simple style of conical hat originating in East, South, and Southeast Asia; and notable in modern-day nations and regions of Bangladesh, Bhutan, China, Cambodia, India, Indonesia, Japan, Korea, Laos, Malaysia, Myanmar, Nepal, Philippines, parts of Russian Manchuria and Vietnam. It is kept on the head by a cloth (often silk) chin strap.
A gugel was a type of hood with a trailing point, popularly worn in medieval Germany. It was tailored to fit the head and shoulders, and was usually made from wool or loden. Originally worn by commoners, it became fashionable with the nobility from the 14th century. In the fashionable style, the gugel was worn on top of the head like a hat, with the head-part inverted inside the collar, which then hung over the ears.
The Phrygian cap or Liberty cap is a soft conical cap with the top pulled forward, worn by the inhabitants of Phrygia, a region of central Anatolia in antiquity.
Don't get caught wearing a dunce cap? A dunce is a person considered incapable of learning. Dunces are often comedically shown wearing paper cone hats, known as dunce caps with the word "dunce" or "dumb", or simply a capital D, on them. Schoolchildren were sometimes compelled to wear a dunce cap and to sit on a stool in the corner as a form of humiliating punishment for misbehaving or for failing to demonstrate that they had properly performed their studies.
An alien couple known as Beldar and Prymatt with cone-shaped heads from the planet called Remulak is mistakenly ditched on earth, during a scouting mission for planet conquest. The coneheads would look fabulous in a well designed conical hat.
The headgear of most self-respecting witches and wizards is usually in the shape of a cone.