Paris, like many other big cities, has a bunch of nicknames. Let's look at some of them!
You probably know Paris as "the City of Lights" ("la Ville des Lumières", although in French, one would rather say "la Ville Lumière") but do you know why it is called this way? Well, it's because Paris was one of the first city to be electrified. As good an explanation as any, I guess. Lyon, another big city in France, has the same nickname for probably the same reason.
The most used nickname for Paris is "Paname", which comes from Panama of the canal and hat fame. Originally, at the beginning of the twentieth century, it was slang but nowadays, it's kind of hip to refer to Paris as "Paname". You hear it a lot in French songs (for example, tunes by French singer Renaud).
Another nickname for Paris is "Pantruche". It used to refer to Pantin, a town/city just outside of Paris proper. It's not very common, to be honest, but it's kind of a fun word.
There's also "Babylone" but you don't hear it as often as "Paname" and it's probably used for other cities around the world.
Finally, what's the nickname for a Parisian? It's "Parigot"! Here's a very childish expression that relates to "Parigot": "Parisien, tête de chien, Parigot, tête de veau" ("Parisian, dog's head, Parigot, calf's head". A "veau" is a young cow (animal corresponding to veal). It's clearly derogatory and you would probably hear that expression "en Province" (outside Paris) when the Parisian lifestyle is evoked.