Pumpkinhead is the name given to a revenge demon (or demons), which takes a very supernaturally mutated form of a human, invoked to take painful revenge on those who the summoner commands at the price of their soul. Pumpkinhead can seemingly only be invoked by those who are truly pressed to take revenge, at the cost of a small amount of money delivered to the witch who is able to awaken him. Part of the reason for Pumpkinhead's name is the fact he is kept buried in a large mound in a pumpkin patch. Pumpkinhead usually must be unearthed either by the witch or the summoner, and at that point needs a bit of blood from the summoner (as well as an incantation from the witch) to awaken, from that point the demon carries out whatever task he was invoked for.
As this task goes on, the summoner takes on more Pumpkinhead-like qualities, and the demon likewise. None of the films have shown what happens when Pumpkinhead fully completes his task, as the films' summoners have all died before the revenge is complete. Only the willing death of the summoner will stop the demon in this case, causing the Pumpkinhead to die, and the summoner to become a new demon, having his/her soul stuck in limbo. If the summoner is killed by someone else, the Pumpkinhead will come after remaining victims the next time it is summoned.
The Pumpkinhead can be brought forth by multiple summoners, in such cases the Pumpkinhead will be stronger and larger unless some of the multiple summoners are killed, in which case it will shrink, until one is left, leaving the monster at the normal size. If all but one summoner is killed, the final one is able to sacrifice themself, which will bring about normal circumstances.
As a revenge demon, Pumpkinhead's priority is to make its victims have painful deaths. Pumpkinhead is sadistic and will in fact go out of its way to make its killings more horrible. Pumpkinhead cannot be distracted from its task, though usually only kills at night, thus taking breaks in its mission. The demon will not kill random people in its way (often seeming to resent this fact when coming across non-targeted humans), unless they try to stop it, in which case he has no qualms with murdering them too.
Since summoners take the form of Pumpkinhead if they kill themself to stop the demon, there are a few Pumpkinheads throughout the franchise. The first Pumpkinhead is unknown, and may simply be the incarnation of a demon (Seeing as it does things later "Pumpkinhead"s have not done, such as speak). When its summoner, Ed Harley, killed himself to stop the thing, the man shriveled into a new creature.
Ed Harley was, and still is stuck in limbo thanks to this. The Pumpkinhead version of Ed was employed in Pumpkinhead: Ashes to Ashes, when it eventually died, and became an untouched corpse. The girl who killed herself to stop the monster became the new Pumpkinhead. A similar situation occurs in Pumpkinhead 4, though in that case, both Pumpkinhead and the summoner were lost down a well.