The history of chess pieces is long and complex. I'll take a bit to describe some of that journey, but then I'll focus on the design we use today: Staunton chessmen. First, know that the name and function of chess pieces has changed significantly over the past 1500 years. Players (and even governments) have modified the names of pieces and the rules for various reasons. I describe some of that in previous answers (queen).
Therefore, piece design has first changed to reflect what they're called in various languages. They've also been heavily influenced by the culture which produces them. The most famous example of this is the Lewis Chessmen, whose history and design is the subject of the book Ivory Vikings: The Mystery of the Most Famous Chessmen in the World and the Woman Who Made Them. For a brief overview with pictures, I'd suggest the British Museum's website.
Our current design was adopted in 1849 in London, when Jacques of London produced this design and asked English chess player Howard Staunton to endorse them. We're not quite sure whether John Jacques, the purveyor of the store (which still exists today), or Nathaniel Cook, his brother-in-law, designed the set. But we do know it was heavily inspired by Victorian London (which was itself inspired by Neoclassicism). The knights, for example, were based on the Parthenon Marbles, also on display (with significant controversy) in the British Museum.
Some changes were purportedly made to improve the distinctiveness or durability of each piece. Chess sets during this time sometimes included pieces which were difficult to distinguish from one another and/or top-heavy (e.g. English Barleycorn Chess Set). To address this, the queen in the Staunton set is distinguished by a coronet, not a crown. In addition, each piece has a relatively wide, thick base which can accommodate a lead weight to prevent tipping, a practice continued today, even in many plastic sets.
But these improvements do not fully explain why this set was so readily adopted, and for that we need to turn to its namesake: Howard Staunton. Staunton was likely the strongest chess player in the world at this time. John Jacques asked Staunton for his endorsement, and Staunton liked the set so much that he demanded that tournament organizers adopt the set for events he participated in. The set remained popular long after Staunton retired from competitive chess, and in 1924 the nascent Federation Internationale des Echecs (FIDE) would adopt this set as the official set for all tournaments. Many variants of this chess set have been created for tournaments, World Championships, Olympiads, etc., but the Staunton design remains standard and an overwhelming favorite of tournament players.
The “bishop” was originally meant to be an elephant archer. The pointy hat with the groove started out as stylized depiction of the platform placed on top of the elephant where the archer could shoot arrows from.
