They're from Arabia and they're accumulators.
The Arab oil sheikh is a rich Arab man with big investments in oil. He is typically seen in Arab dress, complete with keffiyeh, sometimes wearing dark aviator sunglasses when traveling abroad, or in a fine western suit and the keffiyeh and sunglasses. He will sometimes attempt to purchase the services of local prostitutes.
Often referred as the source of various pictures of excess luxury in Snopes.com's photography section, claiming they have, for example, cars made of solid silver, or coated in diamonds. (Back when Texans were the ones considered super rich, one example was the Texan who gave away his Cadillac and bought another one once the ashtrays were full. Yeah, and he smoked too, it was that far back.)
This character type is often brought in to teach a character An Aesop about putting his friends before monetary gain. This often involves the Sheikh innocently offering said character a large reward for a task that involves stepping on the character's friends in order to succeed.
A common outcome is that the character fulfills the requirement while managing to help his friends as well, but he then turns down the reward because he's learned that money isn't the only thing in the world.
In the present climate, they might be involved in financing terrorism. Sometimes The Con will involve someone using this trope and posing as a rich Arab to help explain a source of abundant but eccentric money for the mark.
They also turn up a lot in Romance Novels along with older depictions of the Arab Sheikh where they are dark, brooding, passionate and ruling everything he surveys in his desert kingdom with the same tenacity as he takes the woman. Sheikh romance actually gave us the term "bodice ripper" due to the common kidnap-rape-love plots where the Arab can get away with being beyond normal constraints in how he treats the heroine due to his exoticness. He'll still have the education of Lord Nelson though and the manners of a prince which is kind of the point: these books want someone who lives in the closest thing to a modern lavish royal court and acts like the Black Death hasn't gone out of fashion. He will also turn up as a villain trying to buy or kidnap the female lead for his harem.
An indispensable addition to any (generally evil) Cosmopolitan Council. In more comedic works, he is almost inevitably a Funny Foreigner.
In the Norwegian cult classic puppet-movie, Pinchcliffe Grand Prix, visiting Oil Sheikh Ben Redic Phyfasan (a name which in the original Norwegian would rougly translate akin to something like "Sheik Ben Radish Gawd Damn Eet") ends up sponsoring both the construction of the protagonist's Cool Car, Il Tempo Gigante, and the titular Grand Prix. What, exactly, an Arab Oil Sheikh was doing in a tiny Norwegian village on the far edge of nowhere, is never really explored.
He even speaks Broken Norwegian, so maybe he takes a special interest in Norway.
The exact same plot (Arab oil sheik sponsoring race) shows up in the Swedish movie Göta Kanal. Except it's a boat race. And he ends up not paying because his brother launches a coup as he is gone. In the end the Norwegian navy saves the day.
Charles Stross's The Merchant Princes Series continually compares the titular family of worldwalkers to Arab oil sheiks: they're members of a medieval society who've used a natural resource (in their case, the ability to travel between universes) to become wealthy in the modern world.
Hotel Babylon, where a group are seen as enjoying the services of one of the prostitutes that the hotel will get for clients who ask.
An Arab billionaire tried to buy British Airways in one episode of Absolute Power, although he was dressed in more of the way of a traditional businessman. He needed a PR team because he was related to Bin Laden.
One appeared on Alice, offering to marry the sassiest waitress—she was impressed by the over-sized real diamond ring, but was unwilling to be his fourth wife.
A subversion or aversion- in one episode of Minder, the character to be guarded is a wealthy Arab politician who is a rather noble good guy who doesn't show any of the stereotypical love of excess associated with the character. Amusingly, one character in the episode is hired as a temporary butler and believes the stereotype and thus thinks that hiring a white prostitute for his boss is the first thing he should do.
In an episode of America's Next Top Model, a contestant admitted that before the show she had once gone to what she thought was a modelling casting, but turned out to be a dinner with rich middle eastern men trying to solicit young girls to take home as wives.
On 30 Rock, Floyd is up for a big promotion at General Electric. He and Liz Lemon are looking at what must be a multi-million dollar apartment with a river view. Before Floyd works up the courage to commit to it, an Arab Oil Sheikh enters and agrees to buy it on the spot so that his son can store motorcycles in it.
The Muppet Show. One episode has a subplot of Arabs drilling for oil in the guest star dressing room.
In the Omid Djalili show a recurring sketch has a parody of this trope, with a sheikh who keeps striking oil everywhere. Including in a golf course sand bunker and somehow in the middle of a park bench
Another British show, The Real Hustle, used this to get US$80,000 of jewellery from a Las Vegas jeweller. They gave it back.
They used it to steal two cars as well.
Leverage had Hardison do something closely akin to this in "The First David Job".
In the second episode of Round the Twist, Mr Gribble tries to sell Nell's land (which is a day or two away from foreclosure) to a group of Arab sheiks, with the promise of the Twists' lighthouse to follow. When Mr Gribble bad-mouths the sculptures Tony makes, saying "You wouldn't give it to your mother-in-law", one of the sheiks corrects him "mothers-in-law". Later, the sheik sees the sculptures and buys a statue on impulse, seemingly producing $500 from nowhere (when Mr Gribble expresses disbelief, the sheik smiles and says "Give me a hand, Mr Grobble", patting him on the back).
The music video for The Clash's "Rock the Casbah" (which got a lot of play on MTV in the early 1980s) has some fun with this trope as a man who appears to be an Arab Oil Shiekh ends up hanging around with a man who looks like an Orthodox Jew, and the video ends with the two attending a concert The Clash were putting on.
In one episode of Are You Being Served?, an Arab Oil Sheikh visits the store and attempts to buy a pair of trousers in exchange for a goat...When the goat is refused by the sales assistants, the sheik then tries to trade a beautiful woman.
In Highway To Heaven, Mark Gordon (played by Victor French), temporarily pretends to be one of these to fool corrupt businessmen to help Johnathan Smith and Mark Gordon's equally corrupt, but good-natured friends.
In an episode of The Nanny, the titular character visits a Sheikh at his palace in his home country. He of course falls in love with her, and tries to convince her to stay as his wife. The decision Fran has to make isn't exactly made easier by the fact that the Sheikh looks a lot like Mr. Sheffield. Subverted, in that he wasn't actually interested in marriage, but hiring Fran as a nanny for his son and future children. He did seem taken with her, leading her to the wrong conclusion.
Several iterations of the opening credits of Have I Got News for You had a Sheikh floating on an oil gusher in Water Geyser Volley style. In his last appearance, he was shown accompanied by US soldiers; the moment the oil ran out, the soldiers abandoned him in favour of a humble goatherd who had just struck oil.
The Hashishin from Gothic III fit this trope well, although they don't deal with oil, but rather artifacts from excavating ancient ruins.
Fassad aka Yokuba aka Locria in Mother 3. (Well, he looks like one, in any case.)
Doesn't help the fact that the first time you see him, he's in the desert. It also doesn't help with the fact that his shell is located in the Empire Pork Building, with a desert theme.
Maken X and the remake Maken Shao had one. He was one of the villains, and was short and fat, with his stomach stuffed with grotesque mutant tentacles. Did we mention these games were by the same people as Shin Megami Tensei?
On a similar note, a sheik shows up if you initiate a certain line of rumors in Persona 2 and gives you a side-quest dealing with filling out dungeon maps.
In Hitman: Blood Money, one of Agent 47's targets, Mohammad Bin Faisal Al-Khalifa, could be described as an Arab Oil Sheikh. He actually runs a pharmaceutical company, but he's still filthy rich, influential, and dresses in the appropriate manner for such a Trope. For good measure, he owns the Arabian-themed casino where the mission takes place. One of the customers confuses him for one of the wait staff.
Sheikh Salim represents the Middle East in Tropico 4. Salim has many wives and many camels; he will often promise his second-best camel as a gift for meeting certain requests. Should that camel die, he will instead give El Presidente the change his wives fished out of the couch ($30,000).