Morgana is a fairly strong AP mid. So strong, even, that she's usually banned or picked in ranked games. She can deal a fair amount of damage while also being quite resilient herself, and her passive ability Soul Siphon provides her with excellent health sustain. In addition, she has an excellent kit for utility purposes. Overall, a champion worth playing if you can!

A short section explaining what Morgana has to work with, including what I've found to be the best uses for her skills.

Soul Siphon
Free spell vamp! Since you should be using one of your skills to help you farm minions from level 5 onwards, this is pretty much guaranteed bonus sustain in lane. You won't have to spend 1200g on a Hextech Revolver with Morgana. Of course, you can opt to build a Will of the Ancients regardless since the vamp stacks with your passive.

Dark Binding
Your skillshot nuke. It deals excellent damage and snares the opponent for 2-3 seconds depending on its level. If you land it on an enemy champion, try putting your Tormented Shadow underneath them so they'll take at least 3 hits from it before they can move away. Its biggest drawback is that it's slow. You have to either predict where your target will be moving or slow/stun them first in order to reliably land a Dark Binding. It will hit whatever enemy crosses its path, so minions can block it too.
Because of its long range, Dark Binding can also be used to farm if you're being dominated in lane. Woe your mana sustain if you have to do this, though.

Tormented Shadow
The only reason Morgana can push her lane. This skill turns a part of the ground into a poisonous bog of sorts, damaging and reducing the magical resistance of all enemies inside it every second. The magical resistance debuff stacks but disappears after about 1.5-2 seconds if it's not refreshed. The skill can hit up to 5 times, so there's no way you can get 6 or more stacks of this debuff on one enemy. You can use this skill on an enemy that's stunned, snared or otherwise forced to stay where they are to guarantee multiple hits. Also take note of how the first hit of Tormented Shadow occurs the moment you cast it, not after one second!
Tormented Shadow deals enough damage to kill the caster minions when it's level 3 and the melee minions when it's level 4 or 5 and you have some AP, so use it when the minions start attacking and you know they'll stay on the soil for its full duration. This is your main source of farm. Whenever possible, try placing it in such a way that the first hit will also hurt your enemy! Usually, you can place Tormented Shadow so that it also hits (some of) the melee minions.

Black Shield
This skill is AWESOME. Not only does it provide the recipient (you can cast it on yourself or allied champions) receive a substantial shield that absorbs magic damage, they're also protected from any and all forms of CC while the shield lasts! With its 0.7 AP ratio and 355 base shield health when maxed, it is pretty hard to get past for your enemies. Some people might find it annoying that this shield doesn't protect you from physical damage… the way I look at it, people who deal physical damage simply can't use any slow, stun etc. on the person that's shielded until the shield runs out after 5 seconds. Cast it on an ally that's trying to initiate, on your AD carry if they're in danger of being focused, a teammate that's trying to escape from the enemy or on yourself when you need it. Alt+E is your best friend, remember it! Any enemy gank is doomed to fail if you have this beauty available. One final note, if your Black Shield is destroyed by a spell that would normally inflict CC then that CC will still be blocked.

Soul Shackles
A.K.A. Why We Ban Morgana. The main reason my second build contains Zhonya's Hourglass and Rylai's Crystal Scepter. It deals almost as much damage as a Dark Binding upfront and slows the poor saps that were hit while they remain near Morgana. If any of them stay within range (about twice the range for activating the skill) for 3 seconds, they're dealt another hit for the same amount of damage as the first and receive a 1-2 second stun. Use your Black Shield on yourself, walk or flash into the enemy team, ult and use your Zhonya's Hourglass if necessary. If the enemy doesn't seem like they'll (be able to) burst you down, save the Hourglass active for later and try to move about so as to keep the Soul Shackles on as many (high-priority) targets as possible. If you have them available, place your Tormented Shadow under a stunned high-priority target and land your Dark Binding on them when the stun is about to wear off.
Fun tip: Try combining this skill with Amumu's or Galio's ult so enemies are forced to stay in range of the Soul Shackles for the duration.

Because her Dark Binding is hard to land, Morgana will usually be pretty passive in lane. You might be able to kill your enemy laner if your jungler provides some assistance or once you get your ultimate, but focus on farming and not getting pushed out of lane first. Unless your enemy nukes your waves before your Tormented Shadow can kill theirs ( Mordekaiser is good at this), you should be able to roam or teleport to top or bot lane after you reach lvl 6 and provide some ganks, which will probably be your main source of early kills and supports. In teamfights, you should try and get your Soul Shackles latched onto as many foes as possible, using your Dark Binding to stop runners, freeze a melee champion in their tracks or help burst someone down. Tormented Shadow doesn't cover a large enough are to just toss it down somewhere, so just put it under a stunned or snared enemy that's being attacked by your team to make the most out of the debuff.

This build focuses on making sure Morgana hurts. Some might say the Rod of Ages doesn't match that purpose, but since Morgana needs to be bulky to use her Soul Shackles she needs it. Overall, this build should have enough damage to bring down an enemy squishy in one go. I recommend building Morgana like this if you're doing well in your lane (in Morgana's case this generally means you're outfarming your opponent and not getting killed). You'll be a bigger threat to your enemies with this build, and you're also somewhat vulnerable (although not that much). As you can see, the main difference with build 2 is that I chose to build a Deathfire Grasp instead of a Rylai's Crystal Scepter, because you're focusing on damage instead of utility. This build gives you the best odds at stopping an enemy that's starting to get fed, especially if that enemy is also squishy.

Like I mentioned before, this build focuses more on making full use of the utility Morgana has to offer. Being more bulky allows for better chances of surviving while you're trying to get your Soul Shackles stun off, while Rylai's Crystal Scepter makes it even harder for enemies to escape (or catch) you. This build still deals good damage, but the early Chain Vest and Negatron Cloak and very late Rabadon's Deathcap mean you're hitting for less, especially around 20-30 minutes into the game. Banshee's Veil may be replaced by an Abyssal Mask, although having the spell bubble allows you to use your Black Shield on an ally without exposing yourself too much.

The first 6 levels are not up for debate, except if you need your Black Shield against early ganks.
Morgana is dependent on her Tormented Shadow for farming, so you'll want to at least get it to level 3 in order to kill the caster minions with it. I personally prefer maxing it first so that it's also strong enough to kill the melee minions and deals more damage so that your minions don't wind up dealing the essential last blow. I'd advice to try out both options and decide for yourself which is better.
Dark Binding offers an additional 0.25 seconds of CC and 45 damage per level, so if you're after hard damage you can max it after reaching the preferred level of Tormented Soil. Meanwhile, your Black Shield grants 65 more health for each level you train it, so training it can be invaluable against most aggressive AP mids. Obviously, if you're up against an AD character, you should just leave your Black Shield at level 1 for the CC protection and train your Dark Binding.
If you chose not to max your Tormented Shadow for farming purposes, you'll probably want to max it last, unless the rest of your team is so worthless you need every point of damage you can get to carry them.

Self-explanatory. AP and MPen for damage, armor, mres and health for tankiness, mp5 for sustain. Health sustain is less of an issue for Morgana because of her Soul Siphon, Catalyst the Protector and Black Shield, but feel free to buy health potions if you need them. I choose 21/0/9 for more damage or 9/0/21 for reduced cooldowns on Morgana.

Void Staff
Sure, if you need it. This item can replace your Deathfire Grasp, your Abyssal Mask/ Banshee's Veil or even your Rod of Ages in the late game. Take a second to consider which of the three is more important, though: your own resilience, killing off one priority target fast, or doing better damage to the (off)tanks. This item becomes more viable if you don't use Sorcerer's Shoes and Abyssal Mask.

Will of the Ancients
You don't need it, I told you that. Having said that, this item does provide you with even more health sustain and provides wanting teammates with one of the best auras there are. Worth considering, maybe use mres/lvl glyphs and replace your mres item with this.

Boots of Lucidity and other CDR items
Getting to cast spells more often is always nice. CDR is of lower priority for the Offensive Caster build since your first priority is short-term damage. It can be quite useful to have your ultimate on a lower cooldown in the mid game, but it should be up in time for team fights anyways once you reach late game and death timers go up. Consider Morello's Evil Tome if a Dr. Mundo is proving hard to kill.

Moonflare Spellblade/ Mercury Threads
You have a Black Shield, don't you?

Archangel's Staff/ Athene's Unholy Grail
Decent items, I guess. Athene's is a good item all around, replace your mres item with it if you like.

Mejai's Soulstealer
Like it matters what you build if you're getting that many kills and supports as Morgana. Silly little item that only offers straight-up damage amplification.

Lich Bane
If you already have a Deathfire Grasp and still need more burst. Purchase at your own risk.

Nashor's Tooth/ Guinsoo's Rageblade
…Really now?

Well, that's it for now. I might add more chapters if I feel like it or at request. On that note, feel free to rate and comment my guide if you want. I'd be happy to hear whether I've been of help to you.