Morgana is, in my opinion, one of the bests supports in the League at the moment. Having unique tools in her kit, with high damage included, she is the perfect choice if your team needs ap damage and utility. Even if she has a low skill-cap, playing, building and mastering her will require time and practice. And a bit of skill-shot.

Pros

High damage

High utility

Initiation power

Self-sustain

Cons

Immobile

Squishy

Easy to predict

Soul Siphon (Passive)

Morgana is granted 10/15/20% spell vamp. This amount increases every 6 levels.

Very good sustain tool, u will have better effects into midlane as it scales with ap.

Dark binding (Q)

Fires a bolt of dark energy, rooting the first enemy hit for 2/2.25/2.5/2.75/3 seconds and dealing 80/135/190/245/300 (+90% Ability Power) magic damage.

One of your 2 crowd control abilities, use it to lock people down into botane or make sure enemies will stay into ur W area when playing mid.

Tormented Soil (W)

Curses an area for 5 seconds. Enemies on the cursed ground are dealtfrom 16/32/48/64/80 (+22% Ability Power) to 24/48/72/96/120 (+33% Ability Power) magic damage every second based on the enemy's missing Health.

Best ability to max for increased damage, it also deals missing health damage. Maxing this will grant you both wave clear than kill securing, even more great because it triggers your passive for each target hit.

Black Shield (E)

Shields an allied Champion for 5 seconds. The shield absorbs 70/140/210/280/350 (+70% Ability Power) magic damage and prevents disables until it breaks.

One of strongest shields in Lol, even if it blocks only magic damage, it also makes u immune to disable, but way to counter spells like Charm , Steel Tempest , Pyromania
and thats awesome for midlane, but u can even use it on an ally and help him escape to spells like Death Sentence , Rocket Grab , Winter's Bite . Thats why Morgana is such good support.

Soul Shackles (R)

Dark chains latch onto nearby enemy Champions dealing 150/225/300 (+70% Ability Power) magic damage and slowing them by 20% for 3 seconds. After 3 seconds, they are dealt an additional 150/225/300 (+70% Ability Power) magic damage and are stunned for 1.5 seconds.Enemy Champions can break the chains by moving away from Morgana.

This ability its just great, it's a multi-target stun u can't miss, unless they don't walk away from you. It's good for both mid then bot as its great initiation tool and it has a so high ap ratio, i recommend to try combine it with Orianna s Command: Shockwave.

MIDLANE : Be quiet the first levels of the game, be aware from ganks, harass your enemy with the Q, W combo and then push a bit to get mana back from Doran's Ring, after 6 you can start play more aggressive, and when ur enemy is low enough shield ur self with Black Shield , get near in order to use Soul Shackles , then Dark Binding to disable any mobility spell ur enemy has ( Ahri s Spirit Rush ) and when enemy is stunned use Tormented Shadow for the max amount of damage. It replaces basically replaces Ignite. If you are too low and forced under turrets you can avoid going base, just snare everyone that gets close to you ( after the attacks you so he gets the turret shot ) and wait until you passive Soul Siphon will bring u back to a decent amount of health. if enemy chose to dive, u might be even able to get a kill by hitting Dark Binding or activating Soul Shackles .

SUPPORT : Poke a lot with auto-attacks in order to get more gold and bonus damage with Spellthief's Edge , the best way to do that is hitting Dark Binding and then auto-attacking a couple times, all get even more damage because of Expose Weakness we have in masteries. After 6 coordinate with your adc and look for a 2 vs 2 u easily win thanks to the double stun by Soul Shackles . With good reflexes u should be able to shield your self or the adc with Black Shield just before they get hit by any cc. remember anyway that if the shield falls down, ur immunity to cc do it as well. Tormented Shadow , that we max last when supporting, it's to mana hungry and deals almost no damage as u haven't any ap, so avoid spamming it. Don't be afraid to roam to midlane to help your midlaner, u won't loose that much xp because of Inspiration . Finally remember that Ignite does % health damage, so use it on a low enemy to seal his fate, even if she uses Heal , that becomes useless.

After u buy Sightstone and some Vision Ward s together with the Sweeping Lens u have to start thinking about places where to put wards and what wards to use.

Remember always to carry more than one pink ward with you. Pinks are great for Dragon and Baron control, as they allow u to see and destroy every ward enemy will put trying to steal it. When playing midlane pinks are great coordinating to ur jungler, allowing him to bank one side begin sure it isn't warded. When u are botlane always put a pink on the bush in the side of the map u are dominating, and this can be the river bush or the one next to Dragon's place. The locations I've signed on the map are also good points to spot with Sweeping Lens

Kalista

Kalista & Morgana is a natural very powerful combo, as Kalista's ultimate Fate's Call put's Morgana in range to cast Soul Shackles . Morgana is also a ranged support so it's easier for her to trigger the Kalista's W Sentinel passive, resulting in tons of damage.
Kalista also has her Q Pierce thats a skill-shot, it will be easier for her to lend that if enemies are stunned. Kalista's ultimate is also able to save Morgana from sticky situations, as she is still a bit squishy and enemies tend to focus her.

Varus

Varus it an adcarry that relies much on skill-shots, and even if he has his own stuns and slows he is still having troubles at casting his max-charged Q Piercing Arrow at it's max range, thats what bigger than every cc spell he has. If enemies are locked down by Morgana's Dark Binding or Soul Shackles this will give Varus the time to :
Lend his max-charged Q
lend his ultimate Chain of Corruption that will stun the enemy even more
Lend basic attacks that will trigger his Blighted Quiver for lot of % health damage.

Caitlyn

Caitlyn is an adc that exceeds in early game, with her great damage and utility. Even if her passive Headshot gives her empowered basic attacks, her most relevant damage font is her Q Piltover Peacemaker , a skillshot that deals lots of physical damage. Even if the projectile travels pretty fast, the skills hit-box it's lower than it seems, and the delay it has makes it easy to dodge, but here comes Morgana with her Dark Binding , making sure enemies won't.
Also if Morgana hits a Dark Binding at level 4 this will allow Caitlyn to place a Yordle Snap Trap that will stun the enemy just after the snare ends, while Morg and Cait finish the target off. If enemies get also stunned by Morgana's Soul Shackles they won't be able to protect each other from Caitlyn's Ace in the Hole