There are 11 weapons in total, each with a specific Material needed to upgrade that weapon at the Forge in the Bastion.
1. Cael Hammer - Something Heavy
2. Breaker's Bow - Something Stringy
3. Fang Repeater - Something Nasty
4. War Machete - Something Sharp
5. Scrap Musket - Something Coarse
6. Duelling Pistols - Something Greasy
7. Brusher's Pike - Something Pointy
8. Army Carbine - Something Fancy
9. Galleon Mortar - Something Burnt
10. Fire Bellows - Something Foul
11. Calamity Cannon - Something Wrong
These weapons and their materials are scattered throughout the levels, normally in fairly obvious places (the last weapon is found in the 'Zulten's Hollow' level). Each weapon must be upgraded at least once
If you do not have a certain material, many of them can be found as prizes in the Proving Grounds. Also, once you have built the Lost and Found building in the Bastion, most materials can be found there.
Upgrading also requires some of the game currency (small blue shards). Good ways of getting these are completing the Proving Grounds, Who Knows Where levels and by activating idols once you have built the Shrine in the Bastion.