Arithmancer
An Arithmancer is a person who is an expert on Arithmancy. The word is a combination of the Greek words "arithmetike," which means "of numbers," and "manteia," which means "divination." Practitioners of Arithmancy offer a system for determining favorable and unfavorable days for any individual. As a general rule, favorable days are supposed to be those that correspond to a person's character number. An "eight" personality, for example, would be advised to schedule important events, like starting a business or getting married so that they occur on the 8th, 17th or 26th of the month. Since any name or word can be converted to a number, arithmancy is also used to reveal "hidden" kinship among people, places and things - the theory being that words and names that share the same numerical value are related and naturally go together.
Legilimens
A Legilimens is a Wizard who has learned the art of Legilimency, sensing the thoughts of another person. They can easily detect truth or deception and what a person is thinking or remembering. It can be performed nonverbally. A Legilimens can influence a mind and is a very highly skilled wizard. Albus Dumbledore and Severus Snape are both experts at this form of magic. Voldemort is also an accomplished Legilimens.
Occlumens
The skill of Occlumency allows a person to block the efforts of a Legilimens to probe their thoughts and memories. A person who practices this rare skill is called an Occlumens. An Occlumens can shield the mind from invasion and influence of a Legilimens. They can clear the mind, supressing only the thoughts , emotions and memories with a great deal of will power. Severus Snape was very highly skilled as an Occlumens to have gone so long, undetected by Voldemort and being a spy for the Order.
Parselmouth
The ability to speak the language of snakes and to command them is extremely rare. Parseltongue is the language of snakes; to a human who cannot speak it, it sounds like hissing without taking a breath. A speaker of Parseltongue is referred to as a Parselmouth. The ability to speak Parseltongue is something for which Salazar Slytherin was famous (hence his nickname of "Serpent-tongue", and the choice of a serpent as the symbol of Slytherin House). Tom Riddle inherited this ability from Slytherin, and used it to control the basilisk from the Chamber of Secrets. Harry acquired the ability to speak Parseltongue from Voldemort, during Voldemort's attack on the Potters. Harry used the ability unwittingly to speak to a boa constrictor in the zoo, and to a snake conjured by Draco Malfoy at the Duelling Club.
Seer
Another rare gift is the Inner Eye; the ability to prophesize. A wizard with this gift is called a Seer. Trelawney does have some small measure of this talent, but it has surfaced only twice that we know of. She is, however, the great-great-granddaughter of a very famous and very talented Seer, Cassandra Trelawney.