Possessed Weapons are a type of random loot found in Darksiders II that are rare drops. When obtained, this type of weapon will start off fairly unremarkable, often possessing few abilities but bearing decent starting statistics. However, these weapons are seemingly sentient and will literally devour other pieces of equipment that are fed to them. Each piece of equipment fills up an experience bar that, upon filling, levels up the weapon and allows it to grow in power dramatically.
As the weapon levels up, it will grow in base power but, more importantly, it will learn new abilities chosen by the player. At each level up the player has a choice of 1 to 4 abilities to add to the weapon that vary by weapon type and weapon level, allowing the player to customize their weapon to their own playing styles. This gives a player the option to make their hammer even more of a powerhouse than it already was, or boost the critical hit chance on their claws to make the rapid strikes even more devastating.
As the weapon levels up, its appearance can shift and change in shape, giving it a much more ornate, arcane appearance. How in depth the weapon evolution goes is unknown for the moment but it seems to be based on weapon type as well as the abilities given to the weapon.
The abilities the player can choose when leveling a possessed weapon are based on the abilities the sacrificed weapons had. When choosing a weapon to sacrifice, abilities that are transferable will have their text colored, while non-transferable abilities will be gray. Some examples of non-transferable abilities are: Damage, Thorns, Reaper Energy per Kill, Wrath Cost and the (somewhat obvious) experience and Gilt boosts.
When too many different abilities are fed into the possessed weapon during one leveling, the game reduces the number of choices to four.
Abilities of sacrificed weapons do not stack, the boost will be the same amount regardless of how many weapons with a specific ability are sacrificed.
During each level up after the first, some of the abilities already on the weapon will get a slight additional boost (seperate from the boost chosen by the player) On the first level the damage stat is raised along with one of the other abilities, on the second the damage and two other abilities get a boost, and so on. The fifth level seems to break this adding rule by boosting one less ability than the previous level.
Along with the armor types and Wrath Abilities, the player is able to customize possessed weapons to fit their playing style. Because of this and the skill tree, when a player levels their possessed weapon up and chooses their skills, they will have a very unique character tailored to their style of play. 
A very important note on Possessed Weapons, their stats grow even more if you give weapons with the same attributes.
I.e; give a possesed shock weapon another weapon with shock damage and its power grows even more.