Propaganda is about changing public opinion on something for political or other motives and many games aim to that very thing. In my opinion both advertising games and educational games aim to change the way people think and can therefore be lumped together under propaganda games.
Like many things in the world of games Hollywood have already tackled with putting propaganda in their entertainment properties, the most blatant and perhaps comical example of this comes from Wayne’s World:
Games don’t tend to be that blunt or honest, or get nearly as much money for tossing in such blatant advertising. Still, we can look at what;s out there in the world of games.
Here’s a fairly random selection of games that lean to the educational side but can also be considered propaganda because of the message they carry.
Real Lives allows you to simulate what it’s like to live anywhere on the planet.
Save the Silly Earthlings from their climate change.
Phylo a trading card game inspired by Pokemon, instead it’s abut real animals.
Sweatshop a game about the horrible working conditions that too many people on the planet suffer through in order for cheap products to exist.
Urgent Evoke an interactive story about saving the world.
World Without Oil is all about getting players to be more conscious of how oil is used in everything.
The Bail Out Game is a chance to relive the welfare the banks got from the USA during the banker caused bank implosions of a few years ago.
For more explicitly educational games please check out the great list at Games for Change.
Somewhat related stuff:
It’s worth looking at how Angry Birds has been manipulated into propaganda in this post at MetaFilter.