Martial Melee Weapons are large, heavy weapons which require considerable training to use effectively, such as swords, battleaxes, and heavy maces. (Esoteric martial arts weapons would also qualify.) They usually do 1d8 damage if they're one-handed, or 1d10 damage if they're two-handed. People who aren't proficient with these weapons can still pick them up and try to hit people with them, but they suffer a -1 penalty to hit and damage, meaning that they'll usually be better off with simple weapons.
Simple Melee Weapons are simpler, lighter weapons which can be used without much special training, such as knives, clubs, spears, hatchets, and quarterstaves. Clubs and knives do 1d4 damage; two-handed staves, one-handed spears, and hatchets do 1d6 damage; and two-handed spears do 1d8 damage.
The only Martial Ranged Weapons in ATWC are bows. Bows inflict 1d8 damage, and can be fired once per round, even from horseback. Guns may hit harder, but bows have an enormously greater rate of fire, making them the overall superior weapon; the trouble is that it takes ages to learn to use them properly, whereas any fool can pick up a gun. Of course, fighters might carry both, firing their gun first and then switching to their bow rather than bothering to reload. Non-proficient characters can use bows - you're fantasy Mongolians! You probably fired your first bow when you were five! - but they get -1 to hit, only do 1d6 damage, and need one full round to reload between shots.
|See? Sword, gun, and bow! It's historical!|
(NB: Strength bonuses (and penalties) do apply to archery damage. A bigger, stronger archer will draw a bigger, heavier bow.)
|He gonna shoot you dead!|
Mounted Archery: If you know how to use a weapon, you know how to use it on horseback. (In fact, if anything, you might be a bit shaky on how to use it when not on horseback.) The only exception are long guns: you can't use a jezail while riding a horse, and muskets suffer a -2 to-hit penalty. (That's what carbines are for!) Otherwise you suffer no penalties for just being mounted, though the GM may impose penalties for hitting targets whilst riding at high speed.
Mounted Melee: The absolute best melee weapon to have when charging someone on a horse is a spear or lance - provided you know how to use it! If you are proficient with martial weapons, then charging someone with a lance or spear allows you to inflict double damage. If you aren't proficient with martial weapons, then they take normal damage +1 and the spear probably gets wrenched from your grasp.
(For the same reason, though, anyone who hits you with a fixed spear as you're charging into them on a horse gets to inflict double damage on you. So you might wanna shoot those guys instead.)
Having all that extra momentum behind you isn't just good for spears, though. Whacking someone with a sword, mace, battleaxe, or similar weapon as part of a cavalry charge allows you to inflict +2 damage as well.
If you're on a horse, and you're fighting someone on foot, then you can wheel around and charge back at them for every attack, getting this bonus every time. Sucks to be them!