The only major financial institution in Harry Potter's magical society (in the U.K.) is the famous Gringotts bank — almost entirely governed by the Goblin race. The location is hidden, obviously, to prevent a stray muggle from accidentally walking in, in the area covering Diagon Alley and thereabouts. The bank stores money in various forms, as well as treasures, relics, or even extremely dangerous items.
For the most part, Gringotts remains free of political influence, as the rise of the Death Eaters upon Voldemort's second coming doesn't quite affect its daily activities. For instance, when polyjuice Bellatrix Lestrange enters, the goblins actually demand some form of identification without which she would not have been allowed access to her vault.
Established In 1474
As one of the oldest banks (and presumably longest in continuous operation) in the Wizarding World is Gringotts, first established in 1474. Prior to this, it is unclear as to how the magical world dealt with monetary transactions — to be honest, the economics of Harry Potter still contains a lot of plot holes.
For instance, why on earth do they have prime number reductions in currency, such as 29 Knuts to a Sickle, and 17 Sickles to a Galleon? It makes accounting so much more complicated, but then, that's what the Goblins are great at.
Named After Its Founder
One of the most famous Goblins in the medieval era was Gringott, who is best known for the financial institution named after him. Sadly, there is nothing whatsoever known about this character, except that he has been eternally memorialized in a Chocolate Frog Card for this monumental achievement.
Also, there exists a golden sculpture depicting Gringott atop a mound of treasure located around the bank's headquarters in Diagon Alley.
Has Switched Hands Twice
Interestingly, almost as soon as Gringotts was set up, the Ministry of Magic decided to take control of it (possibly by force, given the fractious relationship between Goblins and Wizards at the time.)
This state of affairs continued for about four hundred years, after which the government chose to return the institution to the care of the Goblins. This might have been partially because of the results of the Goblin Rebellion of the mid-18th century, but one can never be sure.
Not Entirely Subject To Magical Laws
Gringotts receives some kind of legal protection from the Ministry of Magic's various laws, explaining why the vaults are maintained even if the respective client is convicted of a crime.
This is rather unlike muggle jurisprudence, in which the state has the right to repossess their assets (to some degree.) However, both the Lestrange and Black vaults are left untouched while their owners are in Azkaban. Further, its singular status in magical economy gives it minting rights, which bring a further level of complextiy, taken care of by the Goblin Liaison Office.
Bill Weasley Is An Employee
Apparently, the oldest Weasley brother has always wanted to be part of Gringotts, a dream fulfilled when he graduates in the seventh year. He works for the Wizarding bank as a "Curse-breaker", who is an individual trained to recover precious artifacts and treasures from tombs or far off places, and deal with the inevitable curses and hexes the treasure's previous owners put on them.
However, after he met Fleur Delacour for the first time, he stopped doing his treasure hunting in far-flung places and took up a job in London so that he could be nearby while she took on a job to 'work on her English'.
There Are Six Known Vaults
The most well known is the Potter Vault (687), which consists of a massive amount of money, the specifics of which have been undisclosed. The Black Vault (711) is similarly packed with wealth, and is later inherited by Harry. Vault 713 is where the Philosopher's Stone is kept in the first book.
The Weasley Vault is woefully bare, which is not surprising, although a bit sad. The Lestrange Vault is the most incredible of all of them, which is obvious why they require a literal dragon to guard it.
Has A System For Exchanging Muggle Money
There's no way any system in the Wizarding World would function without a way to transfer between magical and Muggle currency, given that most magic-users share their localities with their counterparts (not to mention all the Muggle-borns who enter Hogwarts each year.)
Gringotts has a dedicated section for this purpose alone, but the actual task of exchange is not complicated. The Goblins later somehow recirculate the obtained Muggle cash without being discovered.
The Use Of Probity Probes
Probity Probes are physical devices that are employed by the guards at Gringotts — they are long thin pieces of a gold-hued metal that are said to expose any sort of clandestine spellwork. These items are also used by the Hogwarts caretaker, Argus Filch.
Thankfully, when the Golden Trio enter the bank to access Bellatrix Lestrange's vault for the Sword of Gryffindor, Harry successfully "confounds" Voldemort's minions to avoid being probed.
Fortius Quo Fidelius
As with all things in Harry Potter, Gringott's also has a latin-based motto: Fortius Quo Fidelius. These words can be translated to "Strength through loyalty", but what they mean in the context of the banking world is not really known.
If it was originally invented by the Goblins, it could secretly mean that these creatures form a bond with each other against the wizards they share a tenuous peace with. Still, the term "loyalty" in the motto doesn't really apply when considered from a human perspective.
Security Measures Can Get A Little Extreme
The amount of security dedicated to each vault differ based on the value of the contents therein — in other words, basic ones like those belonging to the Potters and the Weasleys just need a simple key to enter. On the other hand, the Flamel vault cannot be unlocked without the magical "biometric data" of a certified employee of the bank.
The Lestrange's, as shown in The Deathly Hallows, is protected by a blind dragon, trained in the most cruel manner to perform their specified task. In addition, most vaults are heavily enchanted with all kinds of complicated hexes.