This, like most French foods, is a quintessential peasant food that, through whatever process it is that idolizes french shit, is thought of as something vaguely patrician. Fish soup is exactly what it sounds like; what poor fishermen and sailors eat when they arrive in port. This particular version is considered Provencale, originating in Marseille and the surrounding area. I guess northern sailors eat different stuff. Maybe whale.
The basic idea is to take as many different kinds of the cheapest fish you can find, and cook them down whole with some veggies until its a mushy slurry, and then strain it finely so you're left with a super savory, fishy broth. I will demonstrate the traditional method of service as well.
I found some french language recipes, and they all basically go like this:
Lots of different fish - only big ones gutted, small ones whole.
Brown the fish bits, add veggies and sweat until clear, simmer the whole mess without lid adding water as it boils out for 1-2 hours, and strain through a very fine strainer. Before straining, you will need to mash it up or blend it. I think I used my stick blender. Apparently, it is important that you re-boil the soup after the straining. Finally, season to taste and add a measure of saffron. Also, salt, pepper, and paprika. Ideally, it should be fishy, a bit salty, and only a bit spicy. Herbs are up to you, but I think dill is often used.
Here is my soup cooking.
The asian market where I now do 94% of my shopping has a great variety of seafood, and for this I used salmon heads ($1 a lb) and a big tray of smelt (tiny little bait fish). Since this isn't nearly the breadth of fish that you would ideally use (generally something like 6-10 different kinds of fish are used) I used some fish stock I made awhile back from tilapia carcasses (and I suck at fileting fish so there was lots of goodness in the stock)
For service, aioli is required. It's darn easy to make yourself. Just whisk up an egg yolk with a little water, and, while whisking, drizzle in plain vegetable oil (it actually tastes kinda funky if you use olive oil, I have found). This will emulsify the oil and water together and make mayonnaise. The amount of oil depends entirely on what consistency you desire; the more you add the thicker it will get. I think they've done experiments and found that a single egg yolk contains all the emusifying power you need to make 20+ gallons of mayonnaise, so dont worry about eggs. It will need a good amount of salt to taste right.
Then, to make it aioli, add a bit of turmeric, and pureed garlic. The easiest way to do this is press a fork down on a plate and rub a whole clove against the tines of the fork, pureeing it. I used about an entire head of garlic for 1 quart of aioli, but this is entirely a matter of taste.
For the soup, make some big croutons by toasting pieces of bread, and spread aioli on them. Set them afloat in a big bowl of soup. Optionally, you can sprinkle with cheese; traditionally, gruyere or something similar (emmenthaler, etc). I only had asiago on hand, but whatever works.