Harold swears an oath to William
(Bayeux Tapestry, about 1077 AD)
Feudalism is a way to govern a large state when the king is poor and not very powerful, and when there aren't good roads so it is hard to communicate with people who live far away at the other end of your kingdom.
The king divided his land into fiefs (FEEFS) and gave them to other rich men or women, who were called dukes or duchesses, or sometimes earls or countesses. In exchange, the duke or duchess had to swear fealty (FEEL-tee) to the king - they had to promise to come help the king whenever he asked them, and to do whatever the king told them to do. Each duke or duchess held a large amount of land, so they divided their fiefs into smaller fiefs, ruled by counts or countesses or barons or baronesses. In exchange, the counts had to swear fealty to their duke or duchess.
These counts still held pretty large amounts of land, so they would divide their fiefs into farms, and give the farms out to their own followers. In exchange, the farmers had to swear fealty to the counts. So the feudal system was like a pyramid with the king on top, then a few dukes, then a lot of counts, and then a lot of farmers at the bottom.
When the king wanted to get an army together, he sent messengers to his dukes telling them to come with a lot of men, and the dukes sent messages to the counts, and the counts sent messages to each village telling men to come serve in the army.
But often the lords would decide that they were as rich as the king, and didn't have to do what he said. When the king told them what to do, they would just ignore him. Sometimes the king was so weak he just had to let that happen. Other times, the king would fight a war to try to force the duke to do what he wanted.
To find out more about the Middle Ages, check out these books from your local library or from Amazon:
How Would You Survive in the Middle Ages, by Fiona MacDonald and David Salariya (1997). Funny pictures and text convey real history about the Middle Ages.
Knights & Castles: 50 Hands-On Activities to Experience the Middle Ages, by Avery Hart and Paul Mantell (1998). Part of a series of good hands-on activities books.
Medieval Life (DK Eyewitness Books 2004). Not a lot of details, but a good place to start.
You Wouldn't Want to Be in a Medieval Dungeon!, by Fiona MacDonald (2003). Funny tone, but real information. My kids and their friends liked it.