First of all and if you haven't already done so, watch this two-part CGP Grey series about what the City of London is. Around two minutes into the first part, Grey gives a brief explanation - it's something to do with London being rich, and William the Conqueror.

After the battle of Hastings in 1066, William spent the last twenty years of his life entrenching his hold over England. In doing so, he had castles built to exercise control over strategically important regions and installed in them newly minted Norman aristocrats, loyal to him. Some enemies were too tricky to defeat and to difficult to control by force, so he bought them off - notably King Svend II of Denmark and the City of London. Svend got a large sum of money, but the City of London got a writ, or charter, still in existence, which confirmed that they could keep their "laws and customs [...] as they were in King Edwards (the Confessor) day". In other words, the City could maintain Anglo-Saxon practices like the strict division between Freemen and slaves, implement their laws through a local council of Aldermen, and just as importantly, keep their own nobility. Between the Charter and the time of the Magna Charta, the City got the right to its own Mayor, the Lord Mayor of London, still the nominal leader of the City.

Gradually, many of the Anglo-Saxon laws became impractical and were abolished - in other words, even in the City you cannot expect to pay Wergild, i.e. a fine, for killing someone. A great many other practices, however, are still in effect - most invented after the Magna Charta. A very incomplete list:

• The right to conduct international finance and trade without any kind of oversight.
• The right, as mentioned briefly above, to elect their own Freemen, Aldermen, Sheriff and Lord Mayor, and to have their own police force.
• The right to several parcels of land, which are paid for yearly with a blunt knife, a sharp knife, six horseshoes and 61 horseshoe nails.
• The right to regulate and conduct several kinds of business within the City, e.g. to inspect the fish at Billingsgate Fish Market, and control over several main bridges over the Thames.

The Livery Companies, which are essentially what remain of medieval guilds, used to have further rights, including the right to pass the tools of their trade (including cattle) across the bridges without tax and the right to carry a drawn sword inside the City. These are now abolished, and the Livery Companies are mostly ceremonial and regulatory entities.