Ruin:
Ability:
Over-Drive - Upon activation, you run at extreme speeds for about 10 seconds.
Weapon:
Gravity Spikes - Upon Activation, you hurl yourself into the air, then bring you, and your gravity spikes down harder than the German Censor Committee.

Out-Rider:
Ability:
Vision Pulse - Upon activation, you send out an AoE Pulse that goes through walls and highlights enemy Specialists
Weapon:
Sparrow - Upon Activation, you pull out a Powerful Compound Bow with a decent amount of arrows that stick to surfaces and enemies. the longer you charge it, the further it goes, and if it hits an enemy, they'll go flying back. This Weapon requires much more accuracy than anything of the other specialist Weapons.

Battery:
Ability:
Kinetic Armor - Upon Activation, you become literally bullet proof for about 8 seconds, or until your armor is depleted. Won't protect your from other specialist weapons, Melee, headshots or Explosions though.
Weapon:
War Machine - Upon Activation, you draw an 8-shot Grenade Launcher, which will explode upon impact with an enemy, or will spinter into a cluster-nade if it hits a wall 1st.

Prophet:
Ability:
Glitch - Upon Activation, you will teleport back to a previous location (the location you were in about 3-4 seconds ago). You also carry damage taken, but will drop any explosives stuck to you, and can extinguish fire from Firebreaks Purifier.  The Catch is though, a blue trail is drawn from your location to where you teleported to. This is also the fastest charging Ability or Weapon in the game.
Weapon:
Tempest - Upon Activation, you draw the Multiplayer equivalent of the wunderwaffe DG-2, and are given 8 potential shots. When shot, if it hits an enemy, they will become an electrical beacon that will chain to nearby enemies after about 2 seconds (up to 8 on Console, as many as are around it on PC).

Nomad:
Ability:
Re-Jack - When you die, you are put into a "final Stand-esque" mode where you have the option to bring yourself back to life.
Weapon:
H.I.V.E - Upon Activation, you are given a pod Launcher with 5 shots that will stick to all surfaces (people included). When triggered by an enemy, it will swarm then, and tear them apart (one of the most satisfying Weapons to get kills with).

FUN FACT - You can earn a H.I.V.E whilst having another H.I.V.E active, so you can potentially have as many pods as you want active at one time, provided you don't die whilst the other H.I.V.E is active.

Seraph:
Ability:
Combat Focus - Upon Activation, you earn Bonus Score towards your Scorestreaks (+100 upon a kill, basically giving you double points). This is great for getting high-end streaks faster (R.A.P.S, G.I. Unit, Mothership, etc.)
Weapon:
Annihilator: Upon Activation, you draw a powerful 6-shot Revolver that can kill multiple enemies with one shot (Provided they're lined up).

Reaper:
Ability:
Psychosis - Upon Activation, you spawn 3 clones of yourself that run around and confuse enemy players (even though the clones act like bots on recruit difficulty).
Weapon:
Scythe - Upon Activation, Reapers arm Peels back to reveal a Death Machine, which can 3-shot at most ranges.
(Fun Fact: This used to have 200 Bullets in it, but has been nerfed repeatedly until it has been left with either 120 or 80 Bullets)

Spectre:
Ability:
Active Camo - Upon Activation, you go 95% invisible, and will stay that until you break stealth by firing, or waiting for it to expire. (This is my personal Favourite Ability, as I run a lot of stealth, and this helps me out immensely)
Weapon:
Ripper - Upon activation, you reveal two razor-sharp blades, that offer increased Melee Range, Melee Speed, and make your Melee's a 1-hit kill for a limited time. (This is by far the most rewarding Weapon in my book, as you chaining together multiple stabs feels so bloody amazing.) This is also the fastest charging Specialist Weapon, and Second fastest charging weapon/Ability in the game, second only to Glitch.

Firebreak:
Ability:
Heat Wave - Upon Activation, you essentially Microwave everyone around you, slowing them to a crawl, allowing for easy kills. This also destroys any minor equipment and Nomads H.I.V.E Pods. Fun Fact, if you enemy is at low health, you can burn them to death with Heat Wave.
Weapon:
Purifier - Upon Activation, you pull out a flamethrower that, once it touches an enemy, they will be guaranteed to die, unless they happen to land in water immediately after catching fire.

Warfighter
Payloads:
CLAW - Upon Activation, you pull out a rapid fire machine gun that fires ricochet rounds, allowing you to fire around corners.
Combat Focus - Exactly the same as Seraph's Version in Black Ops 3.
Overdrive - Exactly the same as Ruin's Version in Black Ops 3.

Traits:
Ping - Much like the perk in Ghosts, where you kill an enemy, and it sets off a ping, showing nearby enemies.
Persistence - Turns your streaks into persistant streaks, meaning you can die, but still earn them, albeit at an increased cost, and you can't earn them again in the same match.
Re-Supply - Allows you to do what Treyarch implemented into their scav packs, which is being able to replenish used tacticals and lethals from scav packs.

Merc
Payloads:
Steel Dragon - Upon Activation, you draw out a laser-gun thing that can lock onto multiple enemies at once, which is pretty neat.
Bull Charge - Upon Activation, you raise a shield in front of you and charge head 1st into the fray, killing anyone you touch instantly.
Reactive Armour - A copy of Battery's Kinetic Armor, except now you can absorb headshots, explosives, melee damage and shots from Combat Rig Payloads.

Traits:
Infusion - Much like the perk from Ghosts, which offered decreased time before you would start regenerating lost health.
Man-at-Arms - Basically bandolier on Steroids, as this allows you to carry maximum ammunition and run at full speed, regardless of weapon.
Shockwave - remember that useless dive-bomb from Advanced Warfare? No? Well, it's back, and better? than ever, kinda *question mark?*

Synaptic
Payloads:
The Equalizers - Upon Activation, your arms turn into two Suppressed SMG's that deal a solid amount of damage, with ammo to boot.
Rewind - This is Prophets glitch ability, except when you activate it, you go back in time, regen any lost ammo and health, and get back into the fight like you were fresh.
Reaper - Upon activation, you get down in all fours, and run around like a headless chook, turning everything in front of you into a bloody red mist.

Traits:
Combat Burst - When you get a kill, you move faster
Rush-Down - your thrust-meter regens faster, and people caught under your jumps will be damaged/catch fire.
Propulsion - Basically allows you to move like Advanced Warfare for one Move.

FTL:
Payloads:
Eraser - Basically the Annihilator, except with 8 shots, and can't get multiple people with one shot.
FTL Jump - Upon activation, it allows you to jump foward at an alarming speed (similar to Genji's Dash in Overwatch)
Phase shift - Upon Activation, it literally puts you into an alternate dimension, until it expires. Other players with this ability are able to go into this dimension and fight you there, provided they have phase shift equipped as well.

Traits:
Super-charge - When you kill someone, the enemy will drop a super-charger for your teams Payload Meters, which can be picked up, obviously.
Perception - Essentially Sixth Sense, except for when someone is looking in your direction.
Powerslide - Essentially Afterburner for sliding, and it allows you to aim whilst sliding.

Stryker
Payloads:
Gravity Vortex Gun - Upon Activation, you draw a gun that literally shoots black holes.
Micro Turret - Upon Activation, you place a small turret, that is extremely weak, but hits hard (it reminds a little of Axton from Borderlands 2)
Centurion - Upon Activation, you basically throw down an EMP in an area of effect.

Traits:
Relay - Placed Equipment will ping your Minimap with nearby enemies to where the equipment landed.
Trophy Drone - Gives you a personal Trophy Drone that will block one incoming explosive, but it regens over time.
Hardened - Placed equipment is tougher to destroy, and your equipment persists through death.

Phantom
Payloads:
Ballista EM3 - Upon Activation, you draw out a Sniper-Rifle that needs to be slightly charged before firing but it can lock onto enemies and can pin them to walls.
Active Camo - Same as Spectre's Active Camo in BO3
Pulsar - Just a new name for Outriders Vision Pulse from BO3

Traits:
Marked Target - When you shoot someone, it cancels health regeneration in them for a few seconds, as well as marking them for your team-mates to see.
Rear-Guard - when this is active, any attack from behind will be negated, giving you a chance to react.
Heightened Senses - Sixth Sense and Awareness for Sniper Rifles.

I will Update with Images later, but it's late for me, and I need sleep. so I'll add them in tomorrow