Weapon upgrades in Dark Souls use various types titanite. This page will serve as a guide for upgrading basic weapons.
There are two ways to upgrade weapons: Reinforcement and Ascension.
Reinforcement is improving upon a weapon's current type. Such as upgrading your Short Sword +3 to Short Sword +4, or Divine Halberd +2 to Divine Halberd +3. It can be done by a blacksmith, or even on your own at a bonfire, after you've bought the Weapon Smithbox. Upgrading costs 200 souls, along with the required ore.
Ascension is just that, ascending your weapon to a new tier. This occurs when changing a +5 (or +10) weapon of any type to +6 (or +11) of the same type, or a basic level of a new path. Examples include changing a Claymore +5 to a Claymore +6 or Magic Claymore.
The type, size and quantity of ore required for reinforcement depends on the level and type of weapon.
Upgrading a basic weapon to +5 will always require 9 Titanite Shards. (1, 1, 2, 2, 3)
Upgrading a +5 weapon to +10 will always require 9 Large Titanite Shards. (1, 1, 2, 2, 3)
Upgrading a +5 weapon to +5 Magic/Divine/Fire will always require 10 Green Titanite Shards. (1, 1, 1, 2, 2, 3)
Upgrading a +10 weapon to +14 will always require 7 Titanite Chunks. (1, 1, 2, 3)
Upgrading a +5 Magic/Divine/Fire weapon to +9 will always require 7 appropriate (Blue/White/Red) chunks. (1, 1, 2, 3)
Upgrading a +5 Magic/Divine/Fire weapon to +4 Enchanted/Occult/Chaos will always require 8 appropriate (Blue/White/Red) chunks. (1, 1, 1, 2, 3)
All final upgrades require a single slab of the appropriate color.
There are 10 upgrade paths. They branch into each other as follows:
Large Ember: Depths, Kitchen Area guarded by second butcher
Divine Ember: Darkroot Garden, past Moonlight Butterfly
Very Large Ember: New Londo Ruins, after draining water, go to bottom floor of first building and take staircase up
Large Divine Ember: Tomb of the Giants, guarded by several Giant Skeletons
Large Magic Ember: Duke's Archives, Area where you first met Seath the Scaleless (after killing him)
Large Flame Ember: Demon Ruins, left path before Demon Firesage fog wall
Dark Ember: Painted World of Ariamis, annex section
Enchanted Ember: Darkroot Basin, clearing guarded by mushroom men
Chaos Ember: Demon Ruins, in middle of lava pool, guarded by several Taurus Demons
Crystal Ember: Duke's Archives, Treasure Room