It is called "beating in." The newcomer has to fight. Most of the time the newcomer doesn't have to win. It's about demonstrating that they want in bad enough to take a beating. Respect is given for how damage they can deal out during the beat-in, but it is not the main point.
In some stories the newcomer explicitly has to defeat the guy he's fighting to get into the gang. [[note]] Which is not very realistic. Usually it is a bunch of guys kicking the newbies ass.[[/note]] This can lead to some Fridge Logic. Gangs that require every newcomer to be tougher than the people already there are going to have lots of management changes.
A subtrope of Initiation Ceremony and Rite of Passage, while the guy being fought is a subtrope of Threshold Guardian. Compare and contrast If You're So Evil, Eat This Kitten, which may involve being initiated with some other test.
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Anime and Manga
In Baccano!, Firo has a ritualised knife fight with Maiza to get into the Camorrista. The winner only has to draw blood (attempting to kill their opponent will mean that they're shot). When he takes a hit, Maiza enthusiastically shows it to the don to confirm that Firo's won. Partly because it would have healed a few moments later and made it seem like he was still in the game.
A minor character from Midnight Nation mentions being put through the version where the entire gang beats up on the new member when he's telling his life story.
Film — Animated
In Aladdin and the King of Thieves, the Forty Thieves have an initiation session called "the challenge." Essentially, if you discover the thieves' secrets, the only way you can escape being killed to ensure silence is to fight and kill one of their toughest members, and if you win, you become a member of the gang. In Aladdin's case, he has to fight Sa'luk, the Dragon with an Agenda (and later Big Bad).
Film — Live-Action
In Desperado, The Dragon mentions to the Big Bad that the dragon's nephew wants to get into the cartel. The Big Bad sees the nephew doing some sparring and announces that he's only interested in letting the nephew in if he can beat someone named Cristos. Cristos turns out to be a tattoo covered badass who believes in Fighting Dirty and has martial arts training. He proceeds to unleash a nasty beatdown on the nephew, including kneecapping the unlucky kid. How did anyone else make it into that cartel?
In The Count of Monte Cristo, Edmond washes up on an island and encounters smugglers. They tell him that one of their members has been condemned to death and so if he wants to live, he'll have to kill the guy in a fight (and conversely, if the condemned kills him, his crime will be forgiven). Edmond takes a third option and defeats the guy and then pleads for his life, winning the guy's Undying Loyalty.
In The New Guy, the hapless protagonist has to beat up someone to get respect in prison, and then does the same thing when he goes into a new school in order to establish a bad boy image there.
No Escape. When Ray Liotta's character is dropped off on the prison island where most of the film is set, he immediately encounters a prison gang that demands that Ray's character engage in such a fight, and they'll take him in if he can survive for 5 minutes. He beats the other guywith one move while his opponent is showing off.
The Gamorreans from Star Wars refuse to work for anyone who can't beat them (which brings up some Fridge Logic since the job they're best qualified for is hired muscle). Jabba the Hutt once offered to take on a dozen all at once, provided all combatants wear blindfolds. Once the blindfolds were on, he had all his other thugs pile in at once. The nine survivors were awed at his might.
Played with in Fight Club, where to join the club, new members are expected to fight on their first day. However, subverte in that it really doesn't seem to matter whether they win. It's all about the fight.
In Consider Phlebas, Horza, one of the protagonists, is picked up by Space Pirates, and is told that there have limited beds and food, and so if he wants a spot on board, he'll have to fight to the death. He's put against a young and cocky crewmember, and while Horza is a skilled mercenary, at this point, he's shapeshifted into an elderly body. He defeats but does not kill his opponent, but the captain kills the opponent- basically, he wanted an excuse to get rid of the guy, and since Horza was the better fighter, he found one.
The Iron Dream: When Feric Jaggar encounters the Black Avengers and wants to join them, he has to pass the Test of Water, the Test of Fire, and the Test of Steel. The Test of Steel means to fight Stag Stopa, the commander of the Black Avengers, until the Stopa is satisfied of the new member's worth.
In a BBC series titled Diamond Geezer, one of the protagonists is a young con who has just entered prison, and the other protagonist does a Batman Gambit to help him be treated well there- he sets it up so the young guy has to engage in a boxing match with a bulky prisoner who is an enforcer of a London Gangster who pretty much runs the place from a Luxury Prison Suite. The young guy doesn't win, but he shows enough spunk that the London Gangster looks at him with respect. Immediately prior to the fight, he's told that if he hadn't participated, he'd be subject to harassment from other prisoners, including Prison Rape.
In one episode of Burn Notice, Michael and co. capture a member of The Mafiya, and in order to get information out of him, Michael pretends to be a fellow captive. In order to earn the man's trust, Michael picks a fight with him, accusing him of being a snitch. Michael has to use a fighting style that someone with that background would know, and has to fight carefully- well enough to win, but not quite so violently that it's so clearly a fight to the death that the other guy is more likely to successfully kill him. Michael wins, and the gangster takes a Defeat Means Friendship attitude after that.
In an episode of Arrested Development, Tobias gets permission to go undercover as a prisoner in order to get background for an acting role, and so he get the full experience, the film loving warden (played by James Lipton) "thoughtfully" puts Tobias in a cell with a violent inmate known as White Power Bill who is sort of "in charge" of the place. Things don't seem to be going well for the effiminate and wimply Tobias until he accidentally does a Hannibal Lecture on White Power Bill and drives him to suicide. After that, Tobias becomes the head of a prison gang calling themselves Friends of Dorothy.
In a late episode of Soap Jodie and Maggie track his Baby Mama and daughter to a kung fu castle somewhere in California. He watches two men fight, one of whom is a giant guy (the same actor who fights Indiana Jones in Raiders before he gets killed by the plane propeller). After the big guy wins his fight, it's Jodie's turn with him. It isn't explicitly stated that he has to beat him to get in, but it's strongly implied. Jodie manages to beat the big guy with a Vulcan nerve pinch.
A variation: In an episode of The Jeffersons Jenny is a journalist and embeds herself with a street gang in order to do an "inside" story about them. One of the gang members she meets is a pledge; he has to participate in a gang fight before he can become a full member. That evening the gang has a fight with another gang, and the pledge is killed during it.
There's an episode of OZ where mafia Don Antonio Nappa has a bad Gut Feeling about one of the new members of the black gang he's allied with, and tells its current leader to put the guy through whatever sort of Initiation Ceremony they have (so that the guy proves himself) or to kick him out. They initiate through one of these, the variation where everyone in the gang attacks the new member and the new guy has to try to survive it. As the guy in question was an undercover cop, Nappa was entirely right to be worried.
Admission to the Clan Military in BattleTech works that way. Referred to "The Blooding", at the end of their training Clan Warrior Caste cadets have to fight a live-fire duel with three established Warriors. If they survive and defeat at least one of the warriors they are accepted into the Caste.
In the "Tuala Morn" setting for Fantasy Hero, a person who wishes to join a king's warband must challenge a current member to a fair and non-lethal fight. Winning doesn't guarantee acceptance (a king can turn down a person who won the fight, but is known to be dishonorable), but losing guarantees rejection.
Warhammer 40K: Among the grueling trials passed by those chosen to become Space Marines (combine every urban legend on Special Forces training, square them, add invasive and semi-safe surgical procedures and make the applicant less than fifteen), several Chapters include fighting against actual Space Marines. The object of course not being to defeat them but see how long they last. If a candidate actually suceeded in killing a centuries-old seven-foot-tall Power Armored killing machine, he'd likely be executed for being aided by Chaos.
In Knights of the Old Republic 2, you/the Jedi Exile have to beat a number of Mandalorians in their practice arena to earn their respect.
In Saints Row, you have to take on a mob of the Saints before you can join. You don't have to win against them, but you'll get a different cutscene if you do.
Digital Devil Saga: The Embryon tribe approach a rival tribe, the Maribel, to ask for an alliance against a third tribe, the Solids. The Maribel leader, Jinana, forces the Embryon to undergo a test within her headquarters, that culminates in a fight against her personal bodyguard, Bat.
In the Neverwinter Nights 2 expansion pack, Mask of the Betrayer, the Ice Troll Berserkers have several initiation rituals new recruits must overcome, including staged fights against their various members. Subverted in that their strongest fighter will only challenge you after you've already been made a member and beaten all their other members, but you do get a nice enchanted sword as a prize if you best him.
The prison variant is used in Escape from Butcher Bay, when Riddick kills Rust, the leader of one of the prison gangs, in order to gain respect and influence with his fellow inmates.
Xenogears: When Fei is thrown into D Block prison, he has to fight the current Battle Champ Rico and four of his subordinates as part of a "Baptism" ceremony.
An inversion takes place with the Great Khans in Fallout: New Vegas. In order to become a member of their gang, you have to survive getting beaten to within an inch of your life. If you ask them to stop at any point, you fail. You are allowed to try as many times as it takes through. One NPC laments that he thinks he's just not fit to make it, as he has tried numerous times already.
Done in the game Sleeping Dogs, where Wei Shen must fight some of the strongest in the Sun On Yee in order to prove himself worthy of joining them.
In The Order of the Stick, Belkar has to fight a barbarian to join the barbarians' guild. (He fights several, because he feels like it.)
In the western Street Fighter cartoon, Ryu and Ken go undercover in infiltrate the Mad Gear gang from Final Fight. They're told they can only get in if they can beat Sodom, who was the katana wielding boss from the second stage of Final Fight. Naturally, they're up to the challenge, but it makes you wonder how all the street punks that you can practically one shot in Final Fight made it into the gang.
An accidental example shows up in the first act of an episode of Mr. Bogus. Ratty and Mole are falling down a sewer pipe, when Ratty accidentally falls onto the leader of a group of biker sewer rats, taking out the burly rat leader. Once the burly rat has been defeated, the trio of biker rats run over to Ratty and accept him as their new leader, with Mole just tagging along for the ride.
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