In Civilization VI and its expansions, a project can be undertaken in the Production queue of a city to provide some sort of benefit which doesn't include creating a permanent feature like a unit, building, or District.
There are now three main types of projects:
- City projects, which are more or less the same as in previous games. They are required to unlock some powerful feature, such as nuclear weapons, or perform some empire-wide task, such as launching spacecraft. They may (usually) be undertaken in all your cities (with the exceptions of the Science Victory projects, which always require an active Spaceport in the city), and usually need be completed only once. Their Production cost is fixed. You should undertake these projects only on a need-to-do basis; otherwise, you will be wasting Production capacity.
- District projects, which are closely related to the new districts and Great People. They convert a city's Production into other kinds of yields while the project is active and (usually) provide points toward earning Great People when it ends. They can be undertaken as many times as a player wants in any city that has the requisite district. The Production cost of district projects scales up from a base price based on the number of technologies and/or civics a player has discovered, but this also means that their overall effectiveness goes up (as the points they produce are a function of their Production cost). District projects are a great way to boost some specific yields for a time, and also to accelerate your progress towards attracting Great People - you should use them strategically.
- Gathering Storm adds a third type of project: competition projects. These become available in all cities only after a resolution to start a Scored Competition in the World Congress has been enacted, and turn a city's Production toward increasing the player's score for the competition. The projects have fixed Production costs and are only unlocked while the relevant competition lasts. You should undertake them if you're interested in winning the relevant competition, but should always consider the other opportunities for increasing your score as well.
The various types of projects and their effects are listed here.