What are the special abilities the unique/champion monsters can have?

Unique and champion monsters will have a combination of randomly determined affixes/abilities, such as "nightmarish" or "waller".

What are all of the possible abilities?

Arcane Enchanted - Monsters summon a purple laser beam that slowly spins and deals arcane damage. With multiple of these going, you're going to be navigating a maze to get through it. When they spawn, they indicate which direction they initially face before fully appearing.

Avenger - Only champions can have this. Whenever you kill one, the others grow in size and hit harder. The last one left will hit the hardest of all.

Desecrator - Enemies create a fiery (in appearance) zone of damage underneath player characters that deals physical damage over time. Damage starts low but ramps up quickly.

Electrified - Enemies spew out lightning in all directions whenever they take damage.

Extra Health - Self-explanatory. The bonus health is significant.

Fast - Self-explanatory. The monsters are considerably faster.

Fire Chains - This appears on champions only. When close enough together, there are chains of fire linking the champions to each other that deal damage over time to players who stand in the way.

Frozen - Enemies will spawn frozen orbs that spin for a while, damaging and slowing players, and then explode. The explosion deals cold damage and freezes you in place for a few seconds.

Frozen Pulse - An orb of ice pulsates and moves around the battlefield, slowing and damaging players near it. After a while it will explode, dealing extra damage.

Health Link - When attacking a champion with this trait, the other champions nearby will take part of the damage. This makes it tough to take down groups, but has no affect if one is isolated.

Horde - This can only appear on rare enemies. They get extra minions.

Illusionist - Enemies can create 4 clones of themselves. The clones have severely lowered damage and health.

Jailer - The monsters can imprison players so they take some damage and can't move for a few seconds.

Knockback - The monsters can knock back the hero a significant distance (a little over one screen length). The player is slowed significantly after the knockback effect for a few seconds.

Missile Dampening - This one seems to be rather rarer than the rest. It can only appear on rare enemies. The leader (but not the minions) constantly has a rather large bubble around itself that moves to remain centered on the enemy. The speed of all ranged projectiles that enter the bubble are greatly decreased, although if they do make it all the way to the target they will still inflict full damage.

Molten - Enemies leave a trail of fire where they walk that deals fire damage over time to players who stand on it. They also explode a few seconds after they die, dealing damage in an area around their corpse.

Mortar - Enemies gain a ranged attack that fires a volley of a few lobbing bombs every couple seconds. There is a minimum attack distance for the mortars, so standing right near the enemy will protect you from this attribute. This is very dangerous for ranged characters.

Nightmarish - Enemies that hit you will cause you to flee for a few seconds.

Orbiter - Enemies create a central electric sphere, with smaller electric spheres orbiting slowly around it at different distances away. All said spheres deal damage if they touch you. This covers a very large area, but none of the sources of damage are that difficult to dodge.

Plagued - Enemies can create giant pools of poison that deal damage over time to players. These pools are quite large and last a long time and can easily cover the entire floorspace of a room.

Poison Enchanted - Marks an area on the ground as poison, which then spreads in 4 directions diagonally and lasts for a few seconds. Standing in the poison deals damage, but it's avoidable if you pay attention.

Reflects Damage - Dealing damage to the enemies deals a bit back to yourself. A static amount of damage is reflected each time you hit the target. It is only active for short periods of time during the fight. There is a noticeable graphic change associated with the Reflect Damage effect while active.

Shielding - Enemies can turn invulnerable for a few seconds at a time. The effect is fairly frequent, with about a 50% uptime of invulnerability.

Teleporter - Enemies can teleport.

Thunderstorm - Enemies create lightning strikes that also emit small waves of electricity.

Vampiric - Enemies gain life when they deal damage. This also counts damage from other attributes. For example, if you stand on the plague zone from a plagued enemy who is also vampiric, they will gain health.

Vortex - Enemies can suck you next to them and damage you from more than a screen length away.

Waller - Enemies can create walls to prevent escape. They are very good at placing the walls, and will often barricade you in with them. Many ranged abilities cannot go over the walls, even things that appear like they should. For example, a Witch Doctor's Firebomb cannot go over the wall. It is worth noting that rares and champions place their walls differently. Rares will never give their minions the waller ability, but their walls will create long hallways with a blocked end, essentially caging you in with the rare. Champions each can only create one wall each, which has the possibility to trap you in a much smaller area because the layout is not predetermined.

Wormhole - Enemies can create two portals. After a short time, players at either location will be teleported to the other location.

Yes, there is a list here

Here's a copy of the list and their short descriptions. You can view more details at the link above

• Arcane Enchanted - Monsters enchanted with Arcane Enchanted trait deal additional arcane damage and have the ability to summon orbs which fire rotating arcane beams

• Avenger - When a Champion group imbued with Avenger arrives, the death of each group member imbues the remaining Champions with added power as they wreak their frenzied vengeance on the Hero

• Desecrator - A monster with Desecrator trait creates a glowing void zone beneath the targeted Hero that

• Electrified - A monster with the Electrified trait has higher lightning resistance, deals additional lightning damage and discharges sparks across the floor in random directions when stuck. When the monster dies it leaves an electric ball that will continue to generate sparks for a short time

• Extra Health - This trait bestows the imbued monster with extra HP

• Fast - This makes monsters run, attack and cast magic faster

• Fire Chains - Champion backs are attached by a fire chain. The Hero suffers damage if he touches or even gets near one of the fire links.

• Frozen - Monster with the Frozen affix can summon frost orbs that grow and eventually explode causing cold damage and applying a Chill and a Freeze debuff

• Health Link - Health Link reduces the amount of damage taken by the champion with that trait by linking his HP to that of all other Health Link champions in the area

• Horde - Horde increases the squad of minions that spawn with a rare monster, crowding the battlefield

• Illusionist - The Illusionist trait imbues a creature with the ability to create dangerous clones of itself

• Invulnerable Minions - Monsters imbued with this trait arrive with a squad of indestructible puppet minions and an expanded heath pool

• Jailer - This trait gives monsters the ability to immobilize the Hero in a conjured prison

• Knockback - Knockback allows monsters to push back the player a short distance with each connecting blow, whether it be ranged or melee

• Missile Dampening - Creatures imbued with the trait generate a reddish sphere around themselves. Any projectile entering this sphere moves 50% slower than normal

• Molten - Monsters that spawn with this trait deal extra fire damage. In addition, they leave trailing pools of lava that deal fire damage over time. Any non-minion monster killed with this trait leaves a time bomb that will explode after 3 seconds, dealing large amounts of fire damage

• Mortar - Monsters with mortar can lob several fireballs that deal splash damage on impact.

• Nightmarish - Nightmarish monsters have a chance to fear the player on hit and can be triggered from both melee and ranged attacks

• Plagued - Monsters enchanted with Plagued leave pools of poisonous green goo around them for 9 seconds

• Reflects Damage - A portion of the damage dealt to the monster is also dealt back at the player

• Shielding - This trait gives the monster a brief period of invulnerability to all forms of damage

• Teleporter - Gives the monster the ability to teleport to a nearby location

• Thorns - When you strike a Thorns-imbued foe with a melee attack, you suffer automatic damage in return

• Waller - Monsters with Waller can summon earthen walls between the player and the monster, or sometimes behind the player to prevent escape

• Vampiric - This trait allows monster to literally feast on the damage they inflict to you by turning it into health for themselves

• Vortex - Grants the ability to forcibly pull the player right next to the monster

As of patch 2.0.1. There have been a few affixes added and some re-worked. The new list is below. Also, here is the link to the affixes list.

Patch 2.0.5 introduced the following restrictions.

Champions and Rares can no longer spawn with four of the following monster affixes and the frequency at which they can spawn with three has been reduced

Arcane Enchanted
Desecrator
Fire Chains
Frozen
Frozen Pulse
Molten
Mortar
Orbiter
Plagued
Poison Enchanted
Wormhole

Champions and Rares can no longer spawn with both Plagued and Desecrator at the same time

Here is the affix list, with the new affixes in bold.

• Arcane Enchanted: Arcane enchanted monsters summon rotating beams of arcane energy which deal significant damage.

• Avenger: Monsters with the avenger affix grow in power as other monsters in their group are slain — they can increase in damage, attack speed, total life, and other attributes.

• Desecrator: Desecrator monsters conjure burning pools of hellfire beneath your feet. These pools surge after a few seconds, dealing immense damage if you don't move quickly to avoid them.

• Electrified: Electrified monsters release bolts of electricity when struck, causing lightning damage on contact.

• Extra Health: Monsters with extra health have substantially more Life (and, hence, survivability) than normal for monsters of their type.

• Fast: Fast monsters move and attack much more quickly than their normal counterparts. They excel at chasing you down when you're wounded and use their increased speed to move around the battlefield.

• Fire Chains: These monsters are linked together by chains of fire that burn anything they touch. Killing individual monsters in the group limits the length and mobility of the chain.

• Frozen: Frozen monsters fill the air around them with exploding ice crystals. These explosions deal cold damage, slow, and freeze heroes caught in their blast radius.

• Frozen Pulse: Monsters with the frozen pulse affix summon an icy sphere that deals heavy damage over time while it moves across the battlefield.

• Health Link: Monsters with health link share damage and Life pools. Any damage dealt to an individual is split among all members of a group.

• Horde: Hordes of monsters don't gain any particular abilities, but groups with the horde affix are larger than normal for their type.

• Illusionist: Illusionist monsters create duplicates of themselves; these duplicates can deal damage while obscuring your true target.

• Jailer: Monsters with the jailer affix can encase you in a prison, damaging and temporarily immobilizing you. You are vulnerable to attack while immobilized, but you can escape by using certain defensive skills.

• Knockback: Successful hits from monsters with knockback send you flying away from them, interrupting your attack and slowing your movement.

• Missile Dampening: Missile dampening monsters project an aura that slows your incoming projectiles, making ranged attacks less effective.

• Molten: Molten monsters trail burning lava wherever they walk. Shortly after they die, their bodies explode into high-damage gouts of fire.

• Mortar: Monsters with the mortar affix lob volatile globes of fire that can deal extreme damage to you at range. Move close to avoid their attacks.

• Nightmarish: Successful attacks from nightmarish monsters can cause you to lose control of your character and flee for a few seconds.

• Orbiter: Orbiter monsters are able to cover entire areas in an array of electric balls. These balls electrocute anything they touch.

• Plagued: Plagued monsters leave toxic puddles in their footsteps. Standing in these puddles - even to attack the plagued - will deal increasing amounts of poison damage.

• Poison Enchanted: Poison enchanted monsters summon toxic globules that spread in a cross pattern and leave a damaging trail in their wake.

• Reflect Damage: Monsters with this affix deal a portion of the damage they take back to their attackers. Their reflection power turns on and off over time, but you'll be able to tell that it's active if you keep an eye on them.

• Shielding: Shielding monsters can conjure a temporary magic shield that makes them immune to damage.

• Teleporter: Monsters with the teleporter affix can disappear and reappear in different locations on the screen at will, both to make surprise attacks and to escape.

• Thunderstorm: Monsters with the thunderstorm affix use potent lightning strikes to shock enemies from a distance.

• Vampiric: Vampiric monsters feed off of your injuries. They recover a portion of all damage they deal as Life.

• Vortex: Monsters with the vortex affix can pull you towards them, interrupting your attacks and keeping you dangerously close.

• Waller: Monsters with the waller affix create walls that can trap you or block your path to the monster summoning them. Summoned walls will vanish after a few seconds - if you can afford to wait.

• Wormhole: Monsters with the wormhole affix tear a pair of magical rifts in the battlefield. After several seconds, the wormholes emerge, and all heroes near one wormhole are teleported to the other.