The zombie hands are one of the deadliest weapons of a Zombie horde at close range.
As a horde of zombies closes in on its victim, they are able to use their hands to pin them down & painfully disembowel them as they feast. As with its bite, the zombie's hands make use of superhuman strength, letting it tear at flesh and bone with ease. The strength of a zombie would be similar to a heavy weight lifter due to being unable to feel pain in its hands and being extremely aggressive.
Some zombies have claws, either from enlarging of the nails (similar to vampire claws minus the overwhelming speed and not as strong) or from wearing down of their hands. As their flesh falls off, the bones of a zombie are exposed and chiseled down by wear. This makes the hands harder and sharper than before. Some zombie claws can transfer the Zombie Virus by lacerating their victims.
The main weakness of the zombie hands, however, is that it is most effective when used by an entire horde of zombies. A single zombie won't be able to use them as effectively without a couple other zombies to hold the victim down, especially when that victim is as strong as a Vampire. The simulated fight showed that it takes about 4-7 zombies to slow down or pin down a vampire and about 12 zombies can attack at a time when surrounding a single opponent.
While disemboweling their foes, Zombies will start biting and eating them, delivering more lethal damage if the person hasn't already died.
Zombies, being originally viewed as supernatural monsters, would be superhuman because of their dark origins or based on the fact that they were near-indestructable corpses. Zombies of the past rarely formed hordes, so individually would be stronger than most zombies of today.
Romero's Zombies were more humanlike but made them slightly superhuman in strength so they could break through barriers created by the human survivors. It is common for zombie films to show the zombies break through windows and doors with their strength, being too mentally retarded to simply open them or even attempt to. Anyone near a wall or window to the outside could get grabbed by the zombies breaking in, even if the zombie cannot fit through the hole it made it will still use its hands to pin someone to that wall or door and star pulling or clawing them appart.
Throughout Zombie Film history, the Zombies have tried breaking down many varieties of barriers. Ordinary house windows are easily broken into by a Zombie Horde and the horde can easily attack anyone standing near the windows, which happens in many Zombie media. Wooden barriers can take either a minute to a few minutes for Zombies to break through. Zombies may still try to reach through the gaps between the wooden blanks and attack this way. Wooden doors can be broken through quickly, as seen on the show, but will take longer if reinforced. Mall doors take at least 5 minutes to break through, as door glass is much stronger than window glass. Metal barriers rarely can be broken through, even by a zombie horde, as Zombie's strength and hardness of their hands are still only slightly superhuman. Cars are difficult to break into because of the plastic, though the windows can still get broken.
In Night of the Living Dead, the zombies pulled one of the human characters into the horde where she was killed offscreen. The idea of the Zombie Hands was to have so many zombies holding someone down with their superhuman strength that they couldn't resist. Night of the Living Dead showed the Zombie Hands pull the humans towards the Zombie's head so they could quickly kill the person with a bite to the skull.
The Zombie horde's abillity to disembowel a person was fully shown in Romeroe's Day of The Dead (1985). These zombies aimed for all parts of the body; tearing off a person's face, clawing at people's eyes, ripping limbs off and decapitation by tearing through the neck or pulling off the head. Any person getting disemboweled would be in terrible pain and even if the zombies stopped, the damage to the body would quickly kill the person. While disemboweling a person, some of the zombies will bite off the flesh while most would be pulling it off.
A single Zombie can still attack a person without the need of a horde. With their strength they can strangle, throw or slam a person into the ground. This normally takes a while to kill a person though and would be suicidal against someone who is armed or who isn't alone. If for some reason a zombie doesn't want to kill their target the zombie can easily carry them. Most of these actions were performed by Frankenstein's Monster. Zombies have occasionally gored out eyeballs, rip open mouths, broken jaws and snap necks single-highhandedly.
Some zombies of today like the Tank from Left 4 Dead and Jason from Friday the 13th are stronger and more durable than regular zombies. This could be from the virus increasing the zombie's muscle mass or occult magic enhancing the zombie's body.
It is common to find some zombies missing an arm or leg, either because it rotted off or was damaged in battle. Zombies do not care if their limbs are damaged and some will still use them despite being only stubs. If a zombie has no more legs, they will drag their bodies by their arms.
Sometimes, an amputated zombie arm will act on its own, crawling by using its fingers or elbow joint.
If the zombie is a buried corpse, they will use their hands to dig out of the ground. Most films have the hand stick out for a while before the zombie reveal themselves completley. In this scenario, the buried zombies can grab someone's legs and hold them down as other zombies surround and attack them.
It is extremely rare to see any depiction of a zombie using human weapons due to their feral nature and damaged mental abillities. Normally it would be a unique zombie that somehow retained its intelegence yet is still hostile towards people. They would most likely be the most dangerous of the zombies as their weapon and intelegence is combined with the zombie's superstrength and resistance to pain or damage. Otherwise, normal Romero Zombies would most likely use nearby rocks and bricks as melee weapons against doors. Examples are Jason from Friday the 13th, Bub from Day of the Dead, Las Plagas from Resident Evil 4, and the videogame Plants vs Zombies. Some Zombies cannot use their weapons effectively though. In the 2008 remake of "Day of the Dead", Zombies of the American Army can fire guns but cannot aim and so are not much of a threat otherwise.
To see how quickly & effectively the zombie hands could kill a vampire by removing its heart, a ballistics gel torso was laid on its back on a table for three professional strongmen to tear at. However, as zombie expert Max Brooks pointed out, zombies would not possess the knowledge needed to target a specific organ in a victim's body, and would instead view the victim's whole body as one big homogenous lunch.
To compensate for this, the strongmen were blindfolded so that they couldn't see where the torso's heart was in relation to where their hands were digging at. In this condition the strongmen were able to tear open the torso's chest in 20 to 30 and remove its heart in about 58 seconds. This is only with 3 zombies, in a larger horde the time to kill would be cut down. For instance, 9 zombies would kill a vampire in about 19 seconds.
 (Zombie Hands at end of the video)