History

Good answers here already. I would add that feudal society is characterized by peasants being bound to the land. They aren't allowed to leave except with permission. Unlike in the slave mode of production, peasants cannot be bought or sold individually; only the land they live on can be bought, sold, or seized. They also keep about half of what they produce; the lord and the church take the other half. Contrast with the proletariat under capitalism, which loses 99%+ of the value it produces, spending its wages exclusively at businesses owned by the bourgeoisie.

There seems to be some debate over whether peasants owned their means of production, but I would lean toward saying that they didn't—that they rented from the lords, but with conditions that would look pretty lenient but also draconian by our standards (i.e., you can rent this land for a century at a fixed rate, but you can't leave, and you have to fight by my side during the summer if I ask you to). Rents were almost always paid in kind until the 1300s IIRC.

Feudalism is also an upgrade in the productive forces versus slavery, but it still takes quite a long time to really get going (though slavery took even longer). It begins in what we call western Europe with the fall of the Western Roman Empire, but doesn't really start expanding or getting aggressive until the 11th century via the Crusades, the Norman conquest of Arab Sicily, and the Reconquista in Spain. Feudalism has a completely different ideological superstructure compared to slavery: monotheism instead of polytheism, kings and lords and priests being sanctioned by god rather than being gods themselves (the Pharaoh was considered a god, for instance; Caesar claimed descent from Venus and Roman emperors were often treated like gods). Feudalism also has guilds (sort of like unions, but with a heavy family basis and a lot of child exploitation, correct me if I'm wrong here hexbears), which slavery lacks.

Feudalism's economic base is extremely rural, whereas the slave mode of production actually ends up producing massive thriving cities (Rome, Athens, Memphis) with a lot of infrastructure (Roman highways) and enormous architectural projects (the pyramids, the Coliseum). Feudalism, however, chills out the underclass by granting them some concessions both on Earth and in heaven (where they can go so long as they behave on Earth), so uprisings are less of an issue, at least until feudalism starts to fall apart, turning into capitalism in England and absolutism in Spain and France.

The Hundred Years' War and the Bubonic Plague are caused by feudalism's contradictions and likewise bring about its destruction in England, which begins the transition to agrarian capitalism following these cataclysms. Those lenient rents ended up being a huge boon for some feudal tenants—the first capitalists—who made shitloads of money by enclosing land, driving off peasants, and replacing them with more profitable sheep.

Under feudalism, almost all economic activity is undertaken in the name of use rather than exchange. Peasants were mostly self-sufficient. Markets existed but did not dominate society, as with capitalism—they took place every now and then in towns and were mostly concerned with luxury goods, buying cheap and selling dear, and carrying goods over long distances in order to profit. Capitalism is a whole different beast with its relentless focus on market imperatives: if you own capital, you improve it (improve literally means to make it more profitable) or you get proletarianized.

Modes of production is an aspect of Marxism that I've found extremely useful for writing historical fiction. Something I learned that helped a lot was the fact that different modes of production can coexist with one another. One is always dominant, but others can still be present. Feudalism and capitalism coexisted (and arguably still coexist with regard to the monarchy) in England; the mode of production used in the Confederacy is itself some weird shitstorm of slavery, feudalism, and capitalism all mixed together; capitalism still makes use of slavery in prisons as well as feudalism on the margins of empire (debt slaves in Obama-era Afghanistan for instance, the under-development of places like Africa and the Philippines). Also, techno neo-feudalism is just a buzzword as far as I can tell.

I'm getting this information from Wood's The Origin of Capitalism, Federici's Caliban and the Witch, and Marx's Capital. I'm pretty interested at the moment in medieval Italian city states like Venice, Florence, Pisa, and Genoa. (A historian named Lauro Martines has a really interesting book about them.) At first glance these seem to be using the capitalist mode of production—the term "urban proletariat" is used by historians even when referring to workers in ancient Rome, Venice had a massive factory called the Arsenal which produced shitloads of ships, Florence had the Ciompi Revolt—but if you apply Wood's analysis, it looks like these cities are actually just using a unique sort of flavor of feudalism, since you get rich in these places not necessarily by having a business, but by being connected to the government and by investing in feudal lands. This is much less the case under capitalism. In the cities I've mentioned, profit is also made from buying cheap and selling dear over long distances rather than relentless improvement of the forces of production. Economic imperatives are also less present: workers do not always necessarily need to work for a wage in order to survive, they still have access to the commons and could perhaps return to the countryside, etc.