Family Business

Exactly What It Says on the Tin. Basically, a Family Business is any organization run by people who happen to be related. It could be anything, in any time period, across any setting. There is probably an expectation that the children (particularly the eldest son) will carry on the legacy. Truth in Television: many small "Ma and Pop" stores are family owned and operated (kind of goes with the name). But they seem to be dwindling as a result of competition from larger chains.

Not to be confused with The Family's "Business".

Examples of Family Business include:

Anime and Manga

In Hunter X Hunter, the Zoldyck family is literally a family of assassins. From the great grandfather to the youngest of the bunch.

Film

The Godfather, for the ultimate "family business".
- Mafia business aside, they also ran an import business for olive oil.
The Cabots (father and son) in Reservoir Dogs.
The noodle shop run by Po's adoptive family in Kung Fu Panda. Played for laughs because Po is very obviously adopted.
- Not only that, his great-grandfather won it in a game of mahjong.
In Terry Gilliam's The Adventures of Baron Munchausen, Sally asks where's her brother since his father's theatrical troupe was "Salt & Son" He told him there was no son but it was the tradition to put that.
Used a few times in Atlantis: The Lost Empire. His grandfather's influence led to Milo having a love for Archeology, and Audrey joined her father as a mechanic. It's played for laughs with Vinnie, an Explosives Nut, whose florist parents wanted him to join the family business. He relates that the first time he tried to become a florist he got bored, played around with some explosives, and blew up the Chinese laundromat next door. He does say that he (now that he's ridiculously wealthy) is thinking about opening up the shop again.
In Cassandra's Dream, dramatic events probably wouldn't have unfolded if one of the two brothers had been satisfied to keep on running his father's restaurant... and hadn't agreed to help the uncle with the trouble in his business. They're not actually The Mafia, but the subtext exists.
In Hop, the Easter Bunny is apparently a family business. The central conflict of the film is that the latest heir would rather be a drummer than take up the title.

Literature

Over the Wine Dark Sea : Sostratos and Menedemos are cousins working for a shipping business owned by their respective parents
Thomas Mann's novel Buddenbrooks, describing the slow and painful decline of a merchant dynasty in Lübeck, Germany.
Vorkosigan Saga has a lot to do with family. The Vorkosigan's main business is politics but Lord Mark has a number of side projects. Alys Vorpatril handle's Imperial ceremony as well as maintaining some rich apartment complexes. The Toscane's of Komarr are a great shipping firm. The Arquas are a pirate family from Jackson's Whole that engages in a number of projects. The most honest of these is maintaining a port for passing starships but they also indulge in information brokering and under the table dealmaking in Jackson's Whole. What other things they do is not mentioned.

Live Action TV

Ojai Foods from Brothers and Sisters was founded by William Walker and later run by his son Tommy and daughter Sarah. His brother-in-law, Saul, managed the accounting.
Scavo's, the pizzeria Tom and Lynnette open on Desperate Housewives.
The funeral home in Six Feet Under.
On Barney Miller a 50ish painting contracter came from "[Lastname] and Son." Barney asked him what time his son was going to get there, he responded "I'm the son." It had been his father's business but he retired; the son's son wasn't interested in a career in painting, he wanted to be a musician. The son just didn't have the heart to take his old man's name off the company title.
The Bluth Company in Arrested Development.
Albeit unofficially, Keith Mars's private eye business in Veronica Mars.
"Saving people. Hunting things. The family business." In Supernatural, Sam initially doesn't want to carry on the legacy while Dean, the eldest son, does.
The Korean Drama, Twinkle Twinkle is based around the infighting between siblings at a publishing house.
Steptoe and Son: Old Albert and his son Harold. Their scrap business was founded by Albert's father, which actually makes Albert the "Son" in the firm's title.
- And the American version of Steptoe, Sanford and Son. Same deal applies.
Are You Being Served was set in Grace Brothers' department store. The ancient Young Mister Grace put in several appearances; his brother Old Mister Grace didn't get about much.
The business in the Showtime documentary series with the same name as this trope is of course this. Being it's on Showtime, you can probably guess what the Family Business is.
- Hint: The patriarch is better known as Seymour Butts.
Pawn Stars: The Gold & Silver Pawn Shop is run by Rick and his father, "Old Man" and employs his son, "Big Hoss".
The Protector episode "Blood" centered on a family run restaurant operated by a mother, her oldest son and her daughter. The younger son wants to open up a second restaurant under the same name to expand the Family Business but financial and personal issues threaten to derail everything. It Got Worse.
Little House On The Prairie had the Olesons Mercantile and Nellie's Restaurant/Hotel.

Tabletop Games

Traveller: Several really. But an interesting example is the Oberlindes family which brought itself from being a group of Free Traders to near Mega Corp status. They still have something of a style to the way they do business.

Video Games

Jak X: Combat Racing had Rayn swept up into her late father's plot to take over a rival's gang.

Visual Novels

Maji De Watashi Ni Koi Shinasai has several examples, particularly in class 2-S. For instance, the Kuki Financial Conglomerate, exactly as described on the tin. This also gives Ageha, Hideo, and Monshiro rather... distinct personalities.

Web Original

In the Whateley Universe, Goodkind International is really a family business, with Goodkinds working up through the ranks to take over the key positions as older Goodkinds retire or move to the board.
Grandmaster of Theft's Cain International. The protagonist herself is being groomed as inheritor.

Western Animation

In Jimmy Two-Shoes, the Heinous family runs Misery Inc, which exists entirely to Kick the Dog.
Filmation's Ghostbusters: Spenser and Kong, Sr. hand off the business to their respective sons, Eddie and Jake, though they still maintain an active presence.
Sweet Apple Acres in My Little Pony Friendship Is Magic, an apple farm (a giant apple farm) owned by the local branch of the Apple clan.

Real Life

The WWE. Vince McMahon is the chairman, his wife Linda is the CEO, their son Shane is president of new media and global media (i.e. their internet presence and international business), and their daughter Stephanie is the head writer. And it's fully expected that Stephanie will inherit the reins once Vince and Linda retire.
- Just to clarify: That's in real life. Their Kayfabe roles are a little different.
- Don't forget son-in-law Paul Levesque, performer. You may know him as Triple H.
- Shane has since resigned and left the family business.
The English school Dulwich College used to have headmasters with the family name "Alleyn" (after its founder Edward Alleyn). The tradition hasn't been continued for a long time though.
One Real Life example I can think of is Mars (the same one that makes Mars bars and Snickers and all that), which is still private, still family-owned, and based in the one place you'd probably least expect: McLean, VA, just outside Washington DC.
- Candy companies used to be this almost exclusively; now there are only a tiny handful left. Tootsie Roll is another family-owned one.
Another real-life example: S.C. Johnson & Son, the company that makes Windex, Glade and Ziploc.
- Often confused for the above company, Johnson & Johnson is no longer a family business.
And another: Playboy magazine. Hugh Hefner founded it; his daughter currently runs it.
The Adamses, the Roosevelts, the Kennedys, and the Bushes.
Most Kingdoms and Empires in history, often forming a Deadly Decadent Court.
In-N-Out: best damn burgers in the world. Located here in California and owned by family members for generations. Founder set precedent for Bible verses to be printed on the bottom of bags, cups, and cartons. Currently owned by non-family: to be inherited back in years to come.
Members of the Ford family still own the majority of the Ford Motor Company.
McDonald's was once actually run by a real family, the McDonald brothers. They invented the very concept of "fast food" via automatic mechanical cookers and ran the franchise for about 20 years until they partnered up with Ray Kroc, who rewrote all of the signed agreements and took the company over in a hostile takeover, literally leaving the brothers sitting in the desert cooking burgers for random yokels and eternally trying to recapture their Glory Days.
The Whedon family of television screenwriters.
Culver's: A fast growing Wisconsin-based fast casual food chain operating all over the interior US is owned and run by the Culver family.
This is popular among several cultures as an alternative to the more familiar corporate system. It retains a number of advantages including moral cohesion. That is The Patriarch or the Grande Dame probably won't downsize his grandson or granddaughter or even his Old Retainer and in return the later probably won't strike as they are de facto(and possibly de jure as well) shareholders and thus there will be more mutual loyalty and enthusiasm. There is also the matter of convenience. Money can be transferred internally with less red tape, and cousins can be trusted to cover each other's debts thus increasing the credit rating of the clan as a whole.
Adam Bellow wrote the provocatively titled, In Praise of Nepotism. Actually what he meant by nepotism was largely "any networking of social conections, especially blood ties to gain advantage" rather then specifically, "corruption" although he did argue that even that is not totally without advantage and at least there are more distasteful motives for untoward behavior from a person in authority. Nonetheless he felt there were good and bad ways to create and use influence.
Yuri Slezkine in The Jewish Century described this as typical of what he called Mercurian people's, that is ethnic groupings that traditionally focus on maintaining artisanship, commerce, and similar services rather then territory(a characteristic of "Apollonian Peoples"). According to him Mercurians make up for their lack of geographical ties by firm family discipline.
The Chinese have a fondness for "Clan corporations", effectively guilds run under the auspices of a given surname. These provide various services including making business investments, insuring loans by members, educating dependents, etc.
The Rothschilds (who by the way come from the most famous clan of all), have bank branches in several countries. The Central European branch was destroyed by Those Wacky Nazis but several others survive. Aside from banking they maintain oddities like contributions to horticulture and viticulture, as well as patronizing one of the first Zionist settlements.
Travelling peoples, most famously Roma often form family businesses because the most convenient marriages are traveling with the band, or a nearby one not to mention the fact that some practice very strict endogamy. Traditionally such groups provided artisanship and commerce to allow agricultural peoples to continue in their specialties. This practice is less common now then in the past though some Roma certainly spend part of the year travelling and Irish Travellers still maintain a traditional fair.