Gentleman Wizard

"Can a magician kill a man by magic?" Lord Wellington asked Strange.
Strange frowned. He seemed to dislike the question. "I suppose a magician might," he admitted, "but a gentleman never could."

—Jonathan Strange & Mr Norrell

A staple in Gaslamp Fantasy and Fantasy Counterpart Cultures with a Victorian-esque society, is the Gentleman Wizard. He is, essentially, an aristocratic Blue Blood who also happens to be some sort of magician, alchemist, wizard or what have you.

If his magic is something which is passed down through blood, then it's possible he's part of a Magocracy, but he's definitely part of a Magical Society, probably with some interesting name that alludes to hermeticism or Greek Mythology.

If his magic is learned however, then he tends to be close to a magical version of a Gentleman and a Scholar. Either way, his status as a rich blue blood allows for him to get quite good at this magic stuff, since he probably has nothing better to do. Usually the skill to use magic is seen as an appropriate sort of job for a gentleman to have, similar to being a Lawyer or banker. He'll also be a Sharp-Dressed Man, in dapper Victorian or Edwardian attire, possibly embellished to show off the fact that he's magical, and if he has a walking-stick it will no doubt be his wizard's staff or act as a magic focus at least once.

But even as a gentleman, such characters are usually considered to be quite strange and eccentric, even tricky and untrustworthy, no matter how polite their manner may appear to be. And as blue bloods, they might be quite proud and stuffy, looking down on the commoners and the muggles. If magic isn't particularly common then it's not unheard of for him to be The Hermit who lives in a Big Fancy House on a hill which he rarely comes out of, acting as something of an Urban Legend to the populace.

This was the original persona of stage magicians when they first appeared in that era. They have since relaxed as the rest of society has, and now a magician in a suit is seen as old hat.

If he is British, he is most likely a Quintessential British Gentleman; if American, he might be a Southern Gentleman. Compare Gentleman and a Scholar for the science equivalent (the two might dislike each other deeply, but you'd never be able to tell).

Examples of Gentleman Wizard include:

Advertising

The guy wearing the bowler hat with the huge umbrella from the Travellers Insurance commercials a few years ago. Dignified, prim and proper, helping people out using his magic as he comes across them.

Anime and Manga

Mahou Sensei Negima: Negi Springfield, though quite a young one.
England from Axis Powers Hetalia has used magic in the series a couple of times. He's also shown interacting with magical creatures.
Most of the original alchemists in Baccano! appear to be something like this, except for not being aristocrats, especially Szilard and Maiza.
In Cardcaptor Sakura, we have Clow Reed and his reincarnation Eriol.

Comic Books

Roderick Burgess in The Sandman.
Courtney Crumrin's uncle Aloysius.
In the style of Mandrake, there's also Zatara, from DC Comics (best known now as the father of Zatanna).
Doctor Strange, overlapping with Gentleman and a Scholar (since his power is based on his knowledge) and Cultured Badass.

Literature

Most of the magic-users in Mercedes Lackey's Elemental Masters series are this.
Both the eponymous characters in Jonathan Strange & Mr Norrell.
The Chrestomanci are indisputably this in Diana Wynne Jones's Chrestomanci series. Most wizards or magicians in her books follow this pattern.
- Howl from Diana Wynne Jones's Howl's Moving Castle and its Hayao Miyazaki movie adaptation.
Several characters in Sorcery and Cecelia by Patricia C. Wrede and Caroline Stevermer.
From Harry Potter:
- Dumbledore probably counts, especially when he was younger (and had a very stylish looking purple suit when he was visiting Riddle at the orphanage).
- Gilderoy Lockhart wants to be this trope.
- Lucius Malfoy probably was as well, at least until his social standing took a dive after Voldemort's return. In fact, a lot of wizards from the old pureblood families (or at least the rich ones) would probably fit this trope.
Loric from the Temps superhero series.
Dean Henry Fogg of The Magicians makes a deliberate effort to come across this way. One character notes that his speech is so proper, it's almost as though he regretted not having a British accent.
DCI Nightingale from the Rivers of London series, he even has the silver tipped walking stick. And was born in 19th Century too.
In the Gaslamp Fantasy Magician's Ward by Patricia C. Wrede, the protagonist is a young (female) magician who grew up on the mean streets, but has now been adopted by a Gentleman Wizard. At one point she is assured that "a wizard can always be presented [to Society]"—apparently in that version of Regency England, having magical talent automatically allows you entry to the upper class. (But does not excuse poor taste or manners.)
Lord William Beauclerk in the book Bitter Seeds is this trope to a T -- at least at first.
Several minor characters (including a couple of victims) in the Lord Darcy series. Recurring character Lord John Quetzal is an interesting case, as he's a nobleman and a gentleman, but he's from the colonies (Mexico, in our version of reality), which gives him some unusual quirks.
The Wizard in the Land of Oz series is like this, though in both film and books it's obviously an assumed persona for an old carnival ham.
Averted with Uncle Andrew from The Magician's Nephew, who thinks of himself as a gentleman, but rather than being polite and cultured, he thinks it excuses him from such petty restrictions as not tricking an innocent girl into being his unwitting experimental subject.

"But of course you must understand that rules of that sort, however excellent they may be for little boys—and servants—and women—and even people in general, can't possibly be expected to apply to profound students and great thinkers and sages. No, Digory. Men like me, who possess hidden wisdom, are freed from common rules just as we are cut off from common pleasures. Ours, my boy, is a high and lonely destiny."

Wizard Chandler aka "Steed" from The Dresden Files goes to some trouble to appear like one of these. Readers haven't seen enough of him to judge for certain.
- The Merlin of the White Council is also one.
Archchancellor Ridcully in Discworld is an interesting variant; a wizard who is also a country landowner of the huntin', shootin' and fishin' variety.
Felix Harrowgate and other wizards in Doctrine of Labyrinths.

Live-Action TV

Giles could be said to borderline this on Buffy the Vampire Slayer. He's a total British gentleman when you don't make him mad, uses magic although it isn't innate with him, and does used borrowed magic in season 6 to try and stop Willow.

Newspaper Comics

Mandrake the Magician may have been a Trope Codifier, as he was quite the gentleman and quite the mage.

Tabletop Games

Castle Falkenstein: Morrolon definitely counts; indeed, most male sorcerers in this setting do. Most female sorcerers manage to be the Lady Wizard instead.

Theatre

John Wellington Wells, of J. W. Wells & Co a family firm, from Gilbert and Sullivan's The Sorcerer

Video Games

Don Kovak and Vec in Master of the Wind. The former is a Villain with Good Publicity, and the latter is his even more formal and gentlemanly bodyguard. They are also both competent mages. Possibly subverted, since behind the scenes they're quite villainous.