Normal Ascension is performed by Andre of Astora between +5 and +6 and again between +10 and +11.
Andre will require the Large Ember for +5 Weapons and the Very Large Ember for +10 onwards. Once a weapon as been Ascended you can Reinforce your Weapon again with the Weapon Smithbox or simply ask any Blacksmiths for some help. The maximum level is +15.
Armor can be improved up to +10 after Andre has the Large Ember.
The strengths of the weapon are improved on when a weapon is Ascended along the Normal path. These improvements include attribute scaling. Because of this, a +15 Normal weapon may well be the one you find yourself using almost all the time.
A standard weapon at Normal +5 can be modified along different Ascension paths. These are: