Propaganda is the 'special power' of a civic. A propaganda has various effects (good, bad or mixed) and a limited duration.
You can spread a propaganda only after researching its corresponding civic. Spreading a propaganda will cost you an amount of gold depending on the target population. Propaganda success depends on many factors including but not limited to: influence, education coverage, happiness or effects.
If a propaganda doesn't belong to an active civic, it's an inactive propaganda and will have an additional cost in Premium Points*.
All propaganda last for 12 hours.
If you want to be protected from propaganda you can acquire the Sanctuary feature from the Magic Adviser which makes you immune to any propaganda spread onto your city-state for a period of time. This also means you won't be able to spread propaganda on any city-state (including yours) while Sanctuary is active.
* Gold Membership removes the Premium Point cost.