Knucklebones
Knucklebones was played during medieval times, though it actually originated in the days of ancient Greece, with records of the game dating as far back as 330 B.C. The game features four bones, each taken from the ankle of a sheep, with each bone possessing four sides, each of a different shape. Each long side --- convex, sinuous, flat and concave, respectively --- is given a different value, typically 1, 3, 4 and 6. Players roll the bones like dice, and add together their score for the round based on which side of each bone lands facing upwards. If bones aren't available, the fair organizers could substitute them for a set of four-sided dice, available from specialist gaming stores.
Horseshoes
A medieval pastime that's still popular in the 21st century, this game involves throwing horseshoes at a particular target, often a nail. The closer a player gets to the target, the more points she scores, while extra points are sometimes awarded if the player can get the horseshoe wrapped around the target. People have established tournament rules for serious play, available online.
Archery
Archery contests were popular during the medieval era, especially in countries such as England where even the lowest peasants were given plenty of opportunity to train as archers. Participants in an archery contest are given bows and a selection of arrows, and must attempt to hit a target some distance away from them; the closer to the bull's eye of the target the arrow lands, the more points the archer scores. Participants should be carefully supervised by an expert to prevent anyone injuring themselves.
Hammer Throwing
Sportsmen in medieval times threw real hammers around their heads in an archaic version of the Olympic sport. A medieval-themed fair should use a foam hammer for the purpose of the game to avoid unpleasant head injuries. Otherwise, the game is played in the same way as its medieval inspiration: players attempt to throw a hammer farther than their competitors.
Skittles
This game uses skittles much like those found in modern day 10-pin bowling. The skittles are placed at the end of an alley, and players throw wooden balls, typically a little larger than a tennis ball in size, to try and knock as many skittles over during one turn.