The Purpose

Hellboy was originally summoned by the Nazi's as a means to end WWII, his right arm would be the key to doing this. It is unknown how exactly it would accomplish that, but it is known that it can bring and control other greater demons, effectively bringing Armageddon.

Abilities

Hellboy has shown it to be capable for many things in the past. First and foremost is its durability, it is made of a rock-like substances which makes it very hard and impossible to injure. It's also shown mystic properties, such as hitting the hero Ghost (in the Hellboy/Ghost team-up). The full extent of its powers are yet to be realized, but it is implied that it has unharnessed levels of power, that are yet to be discovered.

However, there are also drawbacks. Hellboy cannot load or fire his gun The Good Samaritan because in his right hand because his large rock like fingers don't have the dexterity to properly handle and operate a fiream. The oversized hand also forced Hellboy to become left-handed which has difficulties in itself.

The arm's texture is similair in appearance to Ben Grimm's (The Thing) entire body, with a dark red hue.

The power of the Hand can still be used if separated from Hellboy.

Other Media

In MAD Episode 22 of Season 2: Garfield of My Dreams/I Hate My Teenage Mutant Turtle, Hellboy does a skit where he advertises a product called the Left Hand of Doom. Which is similar to his Right Hand of Doom only for left handed people who can wear it like a glove. Hellboy states that the Left Hand of Doom can gurantee victories in arm wrestling, opening cans of tuna, make left hand turns, and cut paper(which doesn't do the job as implied).