1. Explanations are going to be mostly for arena and will not be Shadowlands specific, so things could be a bit different in the new expansion.  Honestly don't want to get TOO into the unique utility for the sake of lengthiness because this is already going to be very long, so I'm going to be very general for certain things... also with talents/PvP talents you're not necessarily going to have the same set of abilities in every fight.  
2.  
3. **DPS**
4.  
5. *Warrior* - Melee DPS, provides a good pressure when they have uptime.  Good mobility through charges/leaps, but fairly easy to control through roots and other CCs.  Their own CC is fairly limited, in BFA it was an AoE fear on a long CD and a stun.  Lacks self healing, has decent survivability with defensive stance, spell reflect, and plate armor.  The fury specialization does bring strong self healing, but tends to be squishier in return.  Has some decent utility with spell reflect and banner.  Arms is the more popular PvP spec.  Has a mortal strike effect which reduces healing done to the target and has some bleed damage which improves overall pressure.
6.  
7. *Hunter* - Has a pet, two ranged specs and a melee spec that has some ranged abilities as well.  Has quite a bit of control, mostly revolving around a freezing trap every 30 seconds.  Relatively squishy, but has some good tools for avoiding damage.  Utility includes roar of sacrifice which is a defensive CD for yourself or an ally, survival can remove dots from an ally.  BM has more damage tied into their pets, MM has a lot of burst.  
8.  
9. Rogue - Melee DPS with stealth.  Has a lot of CC via Sap, Blind, and stuns with Cheap Shot and Kidney shot.  Thrives on setting up "go" windows where multiple targets are locked down to burst down a kill target.  Built around cooldowns quite a bit so managing them well is a key to survival and landing kills.  Fairly squishy if you're caught with your pants down, but Feint and Crimson Vial are strong for surviving damage you pre-empt, and escaping for restealths is an effective tool defensively as well as allows you to set up goes.  Wound Poison cuts healing done, making it harder to recover during goes. The CC can also be multi use in this regard.  Momentum is pretty big for rogue, when it is in your favor the enemy will have to focus more of their efforts on trying to shut down kills rather than setting up their own.  As well, you want to keep a quick pace to try and outpace the enemy defensive CDs - essentially you land a kill when you get a clean go and the other team lacks an answer.  The better your goes, the quicker the other team will run out of answers.  Assassination focuses on bleeds and consistent damage, though all rogue specs have strong burst as well.  Subtlety has stronger burst and more control.  Outlaw is somewhere in between.
10.  
11. *Mage* - Caster DPS.  Lots of range, lots of control, lots of mobility, strong burst.   Tends to die easily if they get hit, but have lots of tools for avoiding it.  Ice Block is a very powerful defensive that will stop a kill but is on a very long CD.  Has absorb shields on shorter CDs to tide them over.  Polymorph is a spammable CC for setting up kills.  Frost Nova is an AoE root for creating space and controlling melee.  Blink is a medium range teleport, another space creator.  Damage tends to be bursty, requires setup and cross CC for it to really be effective and prevent goes from being shut down.  Frost is focused on their control, lots of slows, fantastic at controlling the flow of the fight and makes playing melee miserable.  Arcane focuses on their arcane spells, better mobility and very high burst if it's set up properly, but more easily shut down because their CC and damage are on the same spell tree.  Fire tends to burn bright and quickly.  Strong burst with a higher pace of play, when they lose momentum things tend to fall apart quickly.  Usually has some good instant damage.  
12.  
13. *Warlock* - Caster DPS.  Good consistent damage, lower mobility, pretty tanky for a cloth wearer, good control, good self healing.  Controls fights through Fear, which forces targets to run away and still allows targets to take a small amount of damage before it breaks.  Demonology is built around pets, requires quite a bit of casting and set up.  Destruction moves more of their damage into casted spells, tends to have high burst.  Affliction specializes in their damage over time effects.  Burst is relatively lower, but the pressure from the DoTs can be overwhelming.  All specs can place a gateway to allow mobility from a set pair of points for themselves and their teammates.  They have some extra CCs to control the fight.  Essentially locks try to control the fight and wear down the enemy.  They can struggle in situations where they are unable to do this and can die if they are overwhelmed.  Lower mobility means that they can be kind of punching bags.  Fighting one can sort of be a race - while you are going for kills they're going to be pushing out control and you'll be taking a beating from their consistent damage, you can end up in situations where the tables turn and you find yourself unable to escape.
14.  
15. *Death Knight* - Melee.   Tanky against casters, strong self healing, strong consistent damage, low mobility.  DKs are like a melee version of a warlock crossed with frost mage, they trade their range for plate armor and damage that's easier to get out. Consistent damage tends to be some of the highest in the game in PvP.  Good at controlling positioning with powerful slows and Death Grip, which is their trade off for the low mobility - rather than getting to your enemy, you're going to make it so the enemy can't move.  Diseases give you some good spread pressure.  Has quite a bit of powerful utility.  Unholy is more focused on their pets and the consistent damage.  Frost tends to be more burst damage and set up burst.   Both have a lot of defensive strength.  Scary situations for DKs are when they can't hit anything for healing or when they are caught out without defensive CDs since they don't really have the mobility to escape.
16.  
17. *Demon Hunter* - Melee DPS.  The most mobile class in the game and a bit of a jack of all trades.  Strong self healing, a powerful defensive CD on a short CD with blur.  Some stuns, an imprison.   Very high consistent damage that goes even higher with CDs.  Has a dispel.  Pretty strong utility.  Weaknesses are that a lot of their effects are not particularly powerful in magnitude compared to other classes with a more narrow toolkit.  Susceptible to being burst down, especially in stuns.
18.  
19. *Monk* - Melee DPS.  Kind of a hybrid of Druid/DH/Rogue.  Has some self healing, strong mobility, good CDs, lots of utility, high burst and consistent pressure, but the pressure requires more uptime and freedom than the DK or DH.  Employs a bit of a hit and run style similar to rogue to offset this.   With CDs available, they move in and pop damage, then avoid damage until things are ready to roll again.  Susceptible to being burst down when CDs aren't available.  The mobility covers good ground, but is not as instant as Rogue or DH, so sometimes they can be easier to control, though they have some tools to mitigate this.  Has quite a bit of soft control rather than the harder CC like rogue.
20.  
21. *Priest* - Caster DPS.  Focuses on damage over time.  Lots of utility and some strong healing.  Strong single target CC for setting up some good CC chains.  Controls the fight through their DoTs and mind control.  Keeps their allies affloat with shields and healing. Has a spammable dispel and is able to defensive dispel with Mass Dispel as well.  The spread pressure can set up their goes.   When things start to dip you hit the healer with a stun/silence combo for guaranteed CC chains. Low mobility but usually has pretty good survivability.
22.  
23. *Shaman* - Caster or Melee DPS. Lots of utility, low CD interrupt, spammable dispel.  Shaman DPS usually has some good burst and consistent damage.  Tends to be less reliable burst than rogue or mage and less consistent damage than warlock or DK.  They excel in the utility department with their low CD interrupt for keeping the enemy out of their comfort zone and totems allowing them to adapt and disrupt the enemy with tools like Grounding Totem and Tremor Totem.   Usually has some pretty good healing.  Elemental is the caster spec.  Has some spread pressure and can burst things down with procs and Earth Shock.  Lightning Lasso can be some reliable burst and lockdown if you are able to to cross CC the enemy team to prevent the other team from stopping the channeled casts.  Enhancement is melee.  Tends to be much more single target focused.  Powerful burst with CDs, but lacks consistent kill potential.  Shamans kind of lack a lot of CC.  Hex is very similar to polymorph, but on a CD so you can't set up the same kinds of chains that Mage can and requires some decision making on how you'll use it.
24.  
25. *Paladin* - Melee.  Plate armor class with lots of utility and healing, very low mobility.  Tends to be very bursty.  High damage with CDs going, pretty low without.  Lacks the slows or control that DK has so it's mobility can make it tough to actually hit targets sometimes.  CC is mostly limited to a long duration dispellable stun.  Has tools like Hand of Freedom for keeping your team moving at full speed.  Blessing of Sanctuary can remove some CCs from your team.  Usually a bit easier to kill than some of the other plate classes, but they tend to provide more support.
26.  
27. *Druid* - Melee or Caster.  Lots of versatility.  Feral is a melee spec that is fairly similar to assassination rogue.  Lots of bleed damage and strong burst with stealth.   Less emphasis on stuns, but has Cyclone and Roots to provide some additional CC.  Has quite a bit of healing.  Employs a hit and run playstyle to cover for the squishiness and take advantage of their healing.  Can swap to bear form to improve their defensives at a great cost to their damage.  Good mobility, but it's mostly based around higher movement speeds and swapping forms to break roots.  Balance is a caster spec with lots of spread pressure through DoTs.  Uses clone and roots to control the fight to a greater degree than Feral does.  Has some high burst.  Adept at kiting, more mobility than priest or warlock, but less than Mage.  Kind of plays like a combination of shadow priest and destruction warlock.
28.  
29. **Healing**
30.  
31. Going to go much less in depth here... tools have quite a bit of overlap with the classes as DPS and general abilities tend to be pretty similar, they just go about things in different flavors.
32.  
33. *Disc Priest* - Lots of focus on damage, damage does some healing.  Tends to have less powerful CDs and on demand throughput, offset by momentum gained through their damage, dispels, and control.  Fares best with high burst go oriented comps.  Not going to touch Holy, this spec has seen quite a few reworks over the past few expansions and I'm not very familiar with how it plays at the moment.
34.  
35. *Druid* - Emphasis on healing over time, usually has good throughput.  The best CC of the healers.  Susceptible to HoTs being dispelled, squishy if caught out of form in a stun.  Tends to fare best with casters and CC heavy comps that enable restealths, drinks, and allow the Druid to roam free.
36.  
37. *Shaman* - Lots of utility, healing has mix of some HoTs and direct heals.  Tends to require casting to get big heals out which at times has made the spec vulnerable to being trained down by enemies with good lock down.  Ideal with partners that have good self sustain allowing the Shaman to prioritize their abilities that aren't healing.
38.  
39. *Monk* - Usually very high output, very high mobility.  Susceptible to CC with poor positioning and squishy.  CC doesn't combo well a lot of the time, but is reliable.  Best with melee comps where their CC doesn't conflict and the strong mobility is an asset or sometimes with casters when the meta allows for their CC to be more or less irrelevant and their output can be leveraged.
40.  
41. *Paladin* - Usually the most CD heavy healing class, survival depends a lot on rotating them correctly.  Often requires casts for significant healing.  High output with CDs but tends to be lower without, sometimes requiring another CD to be used even if freecasting is available.  Lately provides some good burst damage, but you don't want to stay in the position to dish that damage for very long.  Freedom is fantastic to keep one of your DPS mobile.  Tends to be able to play with anything most of the time, but if not overpowered it might not be the absolute ideal healer for a given comp.