Power weapon
Power weapons are a type of weapon in Warhammer 40,000. Power weapons look just like normal melee weapons, but what makes them actually useful is that they are sheathed with an energy field surrounding the blade. This energy field can in turn slice through even the toughest of armor.
For one reason or another, the glorious Daemon Weapons of Chaos operate like power weapons. Probably for the same reason that allows cheap guardsmen bayonets to be exactly as deadly as chainswords in melee combat (that is to say, convenience).
Types of Power weapons
Crozius Arcanum
The signature weapon of Chaplains, the Crozius Arcanum is a mace encased in energy that serves as both a symbol of office and a weapon. Counts as a Power Maul in 6th ed, making the Chaplains suck ass against MEQs but making it concussive like a Thunderhammer.
Power Sword
In a nutshell, the power sword is an example of power weapon technology applied to a sword. It's easily the most common power weapon (Seriously, fuck what Warhammer 40,000: Space Marine says about this, anyone with a functioning set of eyes and access to GW's own site can see it's bullshit), and is used not only by the Space Marines, but by the Commissariat, Inquisition, and Imperial Guard, who, likewise, often issue it to Sergeants and Officers. The Emperor of Mankind and most of the Primarchs who had swords used Power Swords. As of 6th edition, Power Swords are now AP3, screwing over many units and characters who have swords modeled on, such as Mephiston, resulting in Terminators becoming absurdly hard to kill.
A little known fact about Power swords is that if left turned on and dropped, the Power sword will continue to fall and bury itself in the earth until it reaches terminal velocity at the point it passes the planets theoretical centre of mass. It will begin to orbit the centre of the planet hollowing the centre out until its power cell dies. Many Power swords have been lost this way. The sword's hilt would catch the ground and keep it from falling further into the earth. Not to mention the dead-man switches often mentioned being built into them.
Shock Mace
An older-model power weapon favored by the traitor legions, the Shock Mace is functionally a Crozius Arcanum (see above).
Power Axe
Power Axes are single or double-bladed axes with their edges sheathed in energy. Power Axes are especially favored by the Space Wolves and some Chaos Lords.
On the tabletop, Power Axes are AP2, +1S and Unwieldy, making your unit strike at I1. This is probably going to be your only way to cheaply kill terminators in assault, if they haven't overwatched your guy to death or happen to be using Thunder Hammers and Storm Shields.
Power Maul
Power Mauls are whatever form of bashin' stick or stave a Space Marine, human or xenos scum can get their hands on and turn into a relatively hi-tech weapon. If you can conceivably clobber someone over the head with it, it doesn't look heavy enough to be a Thunder Hammer, and it has room to tape a disruption field generator on somewhere along the haft, it's probably a Power Maul.
Crunch-wise, Power Mauls are AP4, +2 Str and Concussive. This means they bounce off MEQs (but wound on a 2+, at least), and when wielded by aforementioned MEQs turn any variety of Spess Pansy or Humie not wearing power armor into meat slurry, multiple wounds be damned. Great for challenges against the aforementioned T3 squishies.
Power Scythes
A form of Power Weapon shaped like a scythe. Power Scythes were generally unwieldy and never caught on with the Imperial military outside the Legiones Astartes, and even then never caught on outside the Death Guard, who used a massive type of Power Scythe known as Manreapers. And even in that case, it was only used by Primarch Mortarion, First Captain Calas Typhon, and the Deathshroud. It should surprise no one that the Departmento Munitorum cancelled all orders for Power Scythes in M34. Whether or not the Adeptus Mechanicus still have the STC printouts are unknown, but knowing the AdMech...
Power Fist
Also known as big-metal-death-fists-of-doom the power fist is one of the most iconic weapons of 40k, Power Fists are massive gauntlets which are encased within an energy field that breaks down molecular bonds when activated. They are intended to be worn with Terminator armour, but frequently you find people wearing Power Armour, or even NOT wearing Power Anything, wielding Power Fists. The Emperor of Mankind had a power fist on his left hand, which he used to famously put Leman Russ in his place. Russ later said that the subsequent headache was from the drinking contest they had.
Orks, in their looting tendencies, have taken to mimicking Power Fists as Power Klaws, which are like Power Fists except that they usually have 2-4 digits, and have massive grabbing blade-claws instead of articulated fingers. Commissar Yarrick famously killed an Ork and used his Power Klaw to replace his severed forearm. Why the Commissariat has not censured him for using xenos weapons is not entirely clear, but they probably let it slide because he's such a badass. Still, one must wonder how he goes to sleep or takes a shower with that thing on.
Some variants of Power Fist are called Power Claws, which are basically Power Fists with the digits ending in razor sharp claws. You more frequently see them among Chaos Space Marines, but occasionally the Raven Guard or Space Wolves use them. The infamous Talon of Horus, used by Horus Lupercal to kill both Sanguinius and the Emperor, and later looted by Ezekyle Abaddon, is a Power Claw with an infamous history. Most of these, however, count as Lightning Claws (see below).
On the tabletop, Power Fists/Klaws double the user's strength, but makes him strike at I1. A popular tactic was to use it to instagib enemy ICs in assault by hiding the Fister behind other expendable troops, allocating wounds onto them instead. However, with 6th edition, this tactic is mostly rendered moot as now ICs and characters can issue challenges, so the Power Fister will most likely fall in battle should the opponent strike and kill him first or see that both get killed off. New wound allocation rules also make the Fister more likely to have a wound allocated to him, but that is slightly mitigated by the Look Out Sir! rule. Dreadnoughts have no such penalty to initiative or benefit of LOS! as they are walking death machines crewed by a crippled vet, yet are strangely not characters.
There has been some speculation on how successful it would be to incorporate Power Fists into truly epic highfives and brofists, but the AdMech has made it perfectly clear that to do so would be tech-heresy.
Lightning Claw
Lightning Claws are a quartet of razor-sharp claws affixed to a gauntlet, each blade of which is a small power weapon. The claws had different designs pre- and post-Heresy, but they mostly function the same.
The pre-heresy design of the Lightning claws is that instead of blades being placed on the dorsal palm, the fingers on the glove are turned into power blades, giving the user a rather terrifying look. This is the wide-spread model that Chaos marines use since it's what they had when they fell back to the Eye of Terror.
The current design of Lightning claws are 3-4 smaller power blades that are placed above the weapon's dorsal palm plating, akin to X-men's Wolverine, which is totally uncool as compared to the old Power Talons.
They are normally used in pairs. Favored by Terminators and Chaos Raptors, Lightning Claws are a bit unwieldy, but are much more likely to land a damaging hit on a target when used by a trained user (in tabletop, this translates to re-rolling failed to-wound rolls). They are also AP3, so it is now pointless to use them against Terminators, but they will mince up MEQs and blobs. The Raven Guard in particular often employ Lightning Claws. Always remember to say "snikt" when using them and to take them off before attempting to masturbate or pick your nose.
Power Lance
A weapon used almost exclusively by the White Scars, Power Lances are energy-sheathed lances which the White Scars use in charging their opponents on their bikes. You can imagine the results.
Tabletop-wise, Power Lances are AP3, +1S on a charge but are AP4 and default strength in following assault phases. Since few box sets come with Power Lance bits, a lot of converting is to be done if you want to kit out your models with Power Lances.
Thunder Hammer
Big fucking hammers that smash anything they touch. Unlike other power weapons, Thunder Hammers only activate their energy fields upon contact with a target. This works like a flash capacitor, and utterly explodes whatever Thunder Hammers hit. Comes in one-handed and two-handed varieties, but both functionally are the same weapon stat-wise. Particularly common among Terminators, although the Salamanders love it (going with Vulkan's training as a blacksmith and the artificer themes of the Promethean Creed). Gameplay wise, they're basically the same Powerfists, except when attacking vehicles they will at a minimum get a "Crew Shaken" if they glance or penetrate, in addition to whatever else they roll on vehicle destruction. In 6th ed, the special effect is changed to the "Concussive" rule; any target that takes a wound (but survives) is reduced to Initiative 1 till the end of the next assault phase, simulating disorientation from the explosive impact, making it very effective even against big gribblies that are tough enough to avoid Instant Death.
Daemonhammer
It's a big thunder hammer that is consecrated with inscriptions, sacred oils, and unguents to allow it to defeat Daemons. In the old Daemonhunters army book, it struck daemons without the initiative penalty a thunder hammer usually had, but as of the Grey Knights codex, it's changed to a Nemesis Force Weapon.
Eldar Power Weapons
The Eldar have the knowledge to create power weapons as well: theirs are of a lighter, more elegant design, and are used almost exclusively by the Howling Banshee aspect warriors.
Eldar Power Sword
Used by the Howling Banshees, these weapons are of a more elegant design than their Imperial counterparts: they have a single edge and have a curve at the tip of the blade. Though being lighter, these weapons have more metal at the tip, allowing them to strike just as hard as an Imperial power weapon.
Mirrorswords
Somewhat smaller than, but otherwise identical to the regular power swords, these blades are used in pairs by the Exarchs of the Howling Banshees. The style that they are used in allows the wielder to deliver a far higher amount of blows on a target than with regular power swords.
Executioner
A glaive, this power weapon greatly enhances the power behind the user's blows, allowing the warrior to slice its enemies in half with ease. Exarchs of the Dire Avengers aspect use a similar, but lighter weapon for superior range over their enemies.
Powerblades
Attached to the user's forearms, these weapons grant the wielder the advantages of a power weapon while allowing the user to keep his hands free. Used mainly by Exarchs of the Warp Spiders aspect.
Scorpion's Claw
Scorpion's Claw is an Eldar version of a power fist, except it doesn't slow its wielder (in game terms it doesn't have the Unwieldy or Special Weapon rules). Striking Scorpion exarchs wear this on one hand while holding a chainsword in the other. It's a big claw with a shuriken pistol inside.
Dark Eldar Power Weapons
While many of the Dark Eldar's weapons have only cosmetic differences to those of Eldar design, there are a few that are more unique
Klaive
These massive two-handed swords are the staple weapon of the Incubus and are regarded by them as "The ultimate weapon". These enormous cleavers are AP2 and +1S while letting you strike at full I. Some squad leaders get the even deadlier Demi-Klaives. They grant an additional S normally, but you can choose to use them as +0S and +2A.
Djin Blade
Only available to the most elite of nobles, Djin blades house the soul of a fallen Eldar. This grants AP3 and +2A, but on a roll of doubles on the extra attacks the soul rebels and the attacks target the user.
Agonizer
A weapon befitting its name, this deadly whip drives the senses of the victim haywire with pain. It grants AP3 and wounds the enemies on a 4+, regardless of toughness.
Huskblade
Carved from Desiccator bone, this horrid saber drains the moisture out of it's victims in a fraction of a second, leaving nothing behind but a shriveled corpse. It grants AP2 and Instant Death