Upgrading your weapon in Dark Souls 2 is done through the blacksmith in Majula. The old hag sells you his key for 1000 souls in the Forest of Giants if his shop is still closed. For a small fee and the materials required he will upgrade your weapon anywhere from +1 to +10. No Embers needed.
Here you can find a quick run-down of all materials used in upgrading your weapons. Remember that the Stone-trader initially found in the Harvest Valley will at some point sell all the materials you need. (A limited supply of Twinkling Titanite, Petrified Dragonbones, and Titanite slabs though). If you want to learn about scalings and base damage increases when infusing or upgrading a weapon I would recommend checking out the Dark Souls 2 Weapon Scalings and Infusion Calculations post.
Used for ascending weapons from unupgraded to +3
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Used for ascending weapons from +3 to +6
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Found in several chests hidden in The Lost Bastille, or The Mines in Harvest Valley
Farmable by killing Dwarfs in Gyrm’s Respite
Used for ascending weapons from +6 to +9
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Used for ascending weapon to +10
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Used for upgrading Boss Soul weapons
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Used for upgrading weapons that can not be normally upgraded
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In Dark Souls 2 you can now imbue your weapon on top of the normal Titanite upgrades. This means that you can turn your +10 Greatsword in a +10 Lightning Greatsword. You will need to visit McDuff the blacksmith in The Lost Bastille and give him a Dull Ember. After you have given him the Ember he will allow you to turn certain Stones into enchantments for your weapons.
Bolt stone
Used for infusing a weapon to deal extra Lightning damage; however, this will decrease all the stat modifiers currently on the weapon, but it will add a Lightning modifier.
Pale stone
You can use the Pale Stone to restore an item by removing its enchantments.
Found on Basilisk in the Shaded Ruins
Raw stone
Used for infusing a weapon to deal extra physical damage; however, this will decrease the stat modifiers.
Firedrake Stone
Used for infusing a weapon to deal extra Fire damage; however, this will decrease all the stat modifiers currently on the weapon, but it will add a Fire modifier.
Poison Stone
Used for infusing a weapon to deal extra Poison damage; however, this will decrease the stat modifiers.
Darknight Stone
Used for infusing a weapon to deal extra Dark damage; however, this will decrease all the stat modifiers currently on the weapon, but it will add a Dark modifier.
Faintstone
Used for infusing a weapon to deal extra Magic damage; however, this will decrease all the stat modifiers currently on the weapon, but it will add an intellect modifier as well.
Bleed Stone
Used for infusing a weapon to apply the bleeding effect; however, this will decrease all the stat modifiers currently on the weapon.