Every time you beat a boss in Dark Souls 2 you're awarded with their soul. You are presented with two choices: do you absorb the soul for a large soul-reward to spend on levelling up and items, or do you save with the prospect of using it at some point later on in the game?
The question is, how do you use these souls to unlock new spells or weapons? The answer lies in a merchant named Straid.
In order to unlock Straid you'll need to have a Fragrant Branch of Yore, which un-petrifies NPCs found in the area that have been turned to stone (there's also a young woman found near Majula suffering from the same ailment that you'll need to cure.
The quickest way to get to Straid is from the Sinner's Rise bonfire. From here, descend the ladder and head back along the bridge towards The Lost Bastille. Once you enter the first room, kill the two mummies. Head up the stairs to the left and kill all the mummies in this room. They can be tricky, so a crossbow or bow is really useful here to avoid alerting them all at once.
Once all the mummies on the top floor are dead, you'll find Straid blocking the entrance to the bonfire in the far cell. Examine him to use the Fragrant Branch of Yore and free him. Speak to him to purchase spells, and trade the souls of fallen bosses (plus a small fee) to gain new weapons and shields. The Smoulder Demon soul, for instance, can be traded for the Smelter Sword, a greatsword that inflicts fire damage (you'll have to pay 1500 souls to get it).
Once Straid is free, he'll remain near this bonfire, and you'll have to kill the mummies each time you return to the bonfire, just be careful not to accidentally hit him during combat.
Before deciding to consume your boss souls, check with Straid as to what weapons and shields you can get for them.