Diplomatic favor is a new type of special currency in Civilization VI: Gathering Storm. It represents your state influence power before other leaders in the world and is used mainly in the World Congress.
Use
It can be said that Diplomatic favor and Influence are the two sides of the same coin: diplomatic power. But while Influence (and the resulting Envoys) represent a more-simple type of diplomatic power, designed to serve with city-states, Diplomatic favor represents the high-level maneuvers used to manipulate that ultimate diplomatic body, the World Congress, where world leaders battle for supremacy.
Diplomatic favor is used to 'purchase' additional votes during every session of the Congress, at the rate of 10 favor per vote. The more Favor a civilization has amassed, and decides to spend, the more votes it will be able to field for a particular Resolution, Treaty or other decision. Given the huge implications certain Resolutions may have for a nation, depending on their current state and goals, the ability to swing a vote in their favor may mean the difference between victory and defeat! And ultimately, each nation will be able to win Diplomatic victory points and head towards the brand new Diplomatic Victory!
Of course, with so much at stake, it is no wonder that leaders start 'trading' Favor as they would commodities or 
Gold. Thus, Trading between leaders is the second main use for Diplomatic Favor. In fact, since the World Congress doesn't start meeting before the Medieval Era, and Trading may happen as soon as you meet a leader, trading Favor well precedes using it for its primary purpose!
Sources
Diplomatic Favor has several stable sources:
- Government. Each type of government provides 1 point per turn, multiplied by the tier of the government. For example, the Monarchy government is of Tier 2, and will provide 1 * 2 = 2 points/turn, while Democracy provides 1 * 3 = 3 points/turn. This is the most reliable source of Diplomatic favor.
- Suzerainty. Being the Suzerain of any City-state provides +1 point/turn.
- Alliance. Having an Alliance provides points per turn depending on the level of Alliance.
You can also win (or lose) Favor by trading with other leaders, as shown above.