The weaponry in Lord of the Rings and Kingdom of Heaven are pretty similar so I've put them in one post.
Chainmail (normal i.e. crusader style): made of crude iron and composed of tiny metal rings. It is handmade. It will stop the point or hack of a sword. All the wearer will get was a bruise. However, it is always rusting. It is expensive. Some men who went to the Crusades sold their own house in order to be able to buy a tunic of chainmail. The chainmail tunic is often worn with a mail hood called a coif. Helmets are not always worn by common foot soldiers but knights wear them over their coifs. Quilted tunics are worn underneath the chainmail to prevent the rings from being driven into flesh.
Swords: Crusaders and Gondorians typically use longswords. These weigh about nine to ten pounds and can be heavier. One blow from a longsword will cut a man from head to sternum. These are made from tempered steel. Due to the weight, it is difficult to maintain a sword fight for more than ten minutes. Rohirrim used swords which are similar to the design of Anglo-Saxon swords. They were about three feet long, double-edged, with slight tapering and rounded point (from
http://www.ancientsites.com/aw/Article/674067).
Horses: For speed, mares are better, but for strength, soldiers rode stallions.
Bows: Longbows are quick and easy to load but are weaker than crossbows. Crossbows take a long time to load (ten seconds). Crossbow bolts can pierce armour. Rohirrim and Muslims use small bows which are easy to use on horseback. Longbow bolts can also pierce armour, but it's more difficult to do so. There have been reports of knights looking like pin-cushions because of longbow bolts but still charging because the arrows just got stuck in their surcoat and didn't pierce the armour.
Shields: Crusader knights carried triangular shields which had rounded edges (think a coat of arms). The shields of Crusader and Gondorian infantry had long shields which came to a point at the bottom. Muslims and Rohirrim had small round shields which were light and easy to use on horseback.
Catapults: There are many types of these. Two of the main ones are trebuchets and mangonels. Trebuchets are the type of catapults which the Gondorians and Balian used. The main part is composed of an 'arm' and a sling where the ammunition is placed. Mangonels are more or less the same, but instead of a sling, they have a bowl at the end of the arm where the ammunition is placed. Typically, the catapult was either powered by men pulling on the other end of the arm, or by a weight at the other end of the arm. For the weight powered catapults, the weight would work the machine when a lever is hit.
Ballistae (sing. Ballista): Basically large crossbows. (In Kingdom of Heaven, Balian used these as harpoons to pull down the Muslims' siege towers).
Leather armour: This is made of stiffened leather. It's lighter than chainmail but also less effective. Good for light armies.
Spears: Rohirrim use ash spears, which are light. They were carried in spear pallets. Muslims have long lances.
Note: I have put the weapons used by Men since there is very little information on the weapons used by dwarves and elves. If anyone finds any, alert me, and I will update this entry.