As far as I gathered from the different prieviews (feel free to correct)
City - works "almost like" your standard Civ city (there are several changes, gameplay-wise, part of them being the management of districts with grow below, the non-allocation of population to tiles, etc...but, in the end, cities keep being the "core" element of your civ)
Town - is a non-operative city (no build queue), having only resource-gathering capabilities. Production gathering is converted to gold and you can still "buy" some buildings that ease town conversion to city at a later stage. Indeed, (and probably except your first), your settlers settle towns, not cities, and you would then decide to evolve them.
Settlements - I cannot recall having heard it fully explained, so I take it is used as a generic for towns + cities. Ah, there is a cap in there, I don't know it would apply only to cities or to all types of settlement.
Districts are anything that expands your city town. It seems now they are generated with population growht and come in two types: rural and urban.
- Rural Distritcts are the replacement for the old improvements, and probably also for citizen allocation to work tiles. This is: now you don't seend your city people out to the wilderness to gather resources, but each time the city grows, you are able to assign a patch of land were they live and work providing you the tile resources (per IGN, you may be limited to adjacent tiles, a la humankind / endless legend, with seems a bad move to me - and to them, but let's wait for more info)
- Urban Districts are also built allocating population and take up a tile (so there is some compromise between "growing" your city or getting resources). Growing your city as urban districts work as building slots. Each urban district will be able to hold a number of buildings (seems it will be increasing with each Era, starting with two in the Ancient), so your will be limited on what you build in your city to the number of Urban Districts you have.
- Quartiers seem to be dedicated urban districts (dedication being generated by the type of buildings you select in them): this is, when you build matching buildings in an urban district, it becomes a Quartier with probably added benefits. The example I have got for this, albeit "unique", explain quite well: as egypt, you can build a Mortuory temple (unique building) and a Mastaba (unique) in an urban district. This will make the urban district become a Necropolis quartier, wich has added benefits over the separated buildings. You may imagine similar potential combinations for "standard" Civ VI districts, so building a library and a school in a district might turn it into a Campus quartier, or a barracks and a guard tower may make a district turn into a Military quartier, etc...