Weapons, Shields and most Armour can be either Reinforced or Ascended. Reinforcement will improve on certain attributes, depending on the Ascension path of the item. Ascension will modify the item, often changing it dramatically.
Demon Titanite is used to Reinforce weapons that have been made from combining an existing Normal +10 weapon and the Soul of Demon.
Anyone can perform Reinforcement at a Bonfire, so long as you have either a Weapon Smithbox or an Armor Smithbox, enough Souls and the Ore required. Alternatively, you can take your item and material to a Blacksmiths who will do it for you - the cost in Souls is the same.
Some weapon upgrade paths are split and require Ascension between points. These points are always multiples of 5. For example, Normal Weapon can be Reinforced to +5, but requires Andre of Astora to Ascend it before you can Reinforce it to +6. This happens again at +10.
Ascending a item is a little more complex, requires a Blacksmith and often an Ember. The Embers are found all over the game world. Each ember can only be taken to a certain Blacksmith (they will let you know if they can use it). And each Ember will allow for a different Ascension path to be unlocked.
The Raw, Demon Soul and Lightning ascension paths do not require any special Embers, but you do require a weapon at the correct Normal level (+5 for Raw, +10 for the others).
No Ember is required for Magic and Fire ascension up to +5, but beyond that you will need the correct Embers.
Andre of Astore, the Blacksmith. Can be found between Undead Parish, Darkroot Garden and Sen's Fortress.
Normal Ascension (Max +15). Improves physical and scaling dmg.
Raw Ascension (Max +5). Greatly improves physical, reduces scaling dmg.
Divine Ascension (Max +10). Adds Magic dmg (Faith scale). Str, Dex scaling reduced. Adds 110 Holy damage.
Occult Ascension (Max +5). Improves Magic dmg (Faith scale). Reduces Str and Dex scaling. Holy changed to Dark.
Magic Ascension (Max +10). Adds Magic dmg (Intelligence scale). Physical dmg, Str, Dex scaling reduced.
Enchanted Ascension (Max +5). Improves Magic dmg (Intelligence scale). Physical dmg, Str, Dex scaling reduced.
Fire Ascension (Max +10). Significant Fire and Physical dmg. All scaling dmg removed.
Chaos Ascension (Max +5). Improves Fire and Physical dmg. No scaling dmg.
Lightning Ascension (Max +5). Significant Lightning dmg added. All scaling dmg removed.
Crystal Ascension (Max +5). Significant increases to dmg. Durability reduced to 1/10 original. Can not be repaired.