Morgana is one of the most all around useful champions. She has the capacity to be built in just about any direction. Most people use her for support, but I find that her natural spell-vamp makes her a lot more tanky than people give her credit for. While support is definitely a great choice for her, you can build her to solo mid and destroy the enemy team. Morgana also has the highest average stats per level, and can theoretically be built for anything from DPS to AD carry. I don't recommend any of those, but it's an option (cue the troll builds).
With a single spell, Morgana can wipe out a whole wave of minions. That spell doesn't even need to be boosted by items, just have a few levels. She can ace with her ultimate, and help protect people who are running from their death, while stopping their impending doom in their tracks. Most games I play with Morgana, I take the middle lane, and destroy turret after turret without any issue.
Morgana's Tormented Soil is a big deal. She has automatic spell-vamp, and an area of effect spell like this will gain you the most health. The coverage area is big enough that if you aim it correctly, you can cover all 6 basic minions. If you have a hard time placing it to cover all 6, you'll want to aim it so that you cover the back 3 minions. Those minions will give you less gold, but you'll have your minions working on the front 3. Also, by aiming at the back 3, you'll also be aiming closer to the enemy champion most of the time. By harassing the enemy champion in this way, you'll be able to keep them away from combat, and keep them from farming gold. A little mentioned side effect of tormented soil is it's effect on magic resist. Enemy champions will have reduced magic resist from this, and make them more susceptible to your spells.
Dark Binding will be another great harass tool. By hitting an enemy champion with this, you will "the unit to the ground, making it unable to move." Note, that this does not disable your enemy. They can still do things, but will generally have a moment of panic when their champion can't move. This is a perfect time to drop your tormented soil under them, and sap more life away. Technically, it would be best to use tormented soil first, and then dark binding after that for maximum damage, it's generally harder to use these two spells in that order effectively. It's better to play it safe and bind, then soil.
Black Shield is an amazing debuff removal tool. It grants protection from spell damage, but the average player will not realize that it does not have any effect against physical damage. Usually you can use this shield to psych out your enemy into thinking you've buffed more than you have. It also prevents stuns, slows, and other abilities that would otherwise impede your movement. If you have already had a debuff like that applied to you, it will remain in effect, so know when to shield yourself, or you'll be stuck fighting for your life when you should be far away from the danger zone. Black shield gives great protection from spell damage early and mid game, especially against champions like Veigar and Karthus.
Soul Shackles is a great ultimate. Not only does it do massive damage, and slow, but if you just watch the icon, you can tell when enemy champions are nearby, even in the bushes. It's great for escaping, slowing enemy champions so you can catch up, or (my favorite) being right in the middle of a team fight, and blasting every enemy champion at once. If your AP is high enough, you can deal massive damage to everyone, throw down a tormented soil to hit 2 or 3 champions at once, and dark bind someone so your team can finish them if you didn't. Mix in your shield for protection, and you've got an absolutely lethal nuke just waiting to happen.
Items for Morgana are going to be pretty common sense until mid to late game. Start with a Doran's Ring to give her a little bit of AP, but mostly you want it for the Magic Regen.
In the mid game, you'll want to focus on cooldowns. Morgana has great stopping power, but the cooldown for her spells is horrible. At max cooldowns, you can cast tormented soil just as soon as the last soil you cast fades away. I generally go straight for the Chalice of Harmony for the magic regen and magic resist. After that, I go for Fiendish Codex for the AP and cooldown. These two items work beautifully together, especially because that puts you 900 gold away from Athene's Unholy Grail which is the perfect item for any mage.
Somewhere before or after getting Athene's Unholy Grail, you'll probably want boots to increase your mobility. Most people take Sorcerer's Shoes for the spell penetration, but I find that Ionian Boots of Lucidity are better for the cooldown reduction they give you. Between the boots, grail, and masteries, you should have yourself maxed out with 40% cooldown. While spell penetration can be really helpful, I generally feel that cranking out more spells is even more important.
After getting the unholy grail, you'll want to continue boosting yourself up. The Abyssal Scepter will grant you 3 special things. First, you get an AP boost. Second, you get more magic resist. Third, you get a special passive that reduces nearby enemy champion magic resist. It's not spell penetration, but it's a good substitute.
After building this far, you'll want to make some situational choices. If you want more AP, then I recommend Rylai's Crystal Scepter. If you really want to tank up (like this build is mostly for), then I recommend Warmog's Armor. Warmog's will give you a great boost to your health, and health regen. If you can get both, you can basically walk up to a turret with no minions, and knock it down by your lonesome.
If you're having a hard time building up, you can take a couple shortcuts. It will end up costing more gold, but a philosopher's stone or Needlessly Large Rod can really come in handy. The philosopher's stone will grant you gold, health regen, and mana regen, which is all around useful for mages going mid-lane. The needlessly large rod can be built into a Rabadon's Deathcap, which will give you a pretty huge boost to AP.
You'll want to start by taking one point in each in this order: Tormented Soil, Dark Binding, Black Shield. After that, you'll mostly want to max out your tormented soil, taking the occasional dark binding when you can't get another tormented soil. Black shield is most important for it's ability to stop slows/stuns, so one is all you really need for most of the game. Obviously, take the point in Soul Shackles whenever you get the chance. Never pass up a chance to level up an ult. For anyone. Ever.
Scaling up your tormented soil is very important. By the time you get 4 points in it, you should be able to wipe out a whole wave of minions in 1 casting. If you bind a champion and put tormented soil under them, you'll reduce them pretty far, especially with the reduction in magic resistance from the soil and the Abyssal Scepter, which makes your spells go even further.
The masteries you'll want to take are going to be spread around a bit.
Firstly, you'll want that 4% cooldown reduction. This 4% will put you at 39% cooldown after you have all of your items selected, which is 1 away from the max of 40. Ability power is also the most fundamental part of any mage. Spellsword will help your auto attacks deal more damage as your AP increases, which will help you spread damage when you're waiting for cooldowns or covering areas that tormented soil isn't hitting.
Not much to do here, except increase your health regen by a max of 6 points per 5 seconds (depending on missing life). It's especially helpful early game when you can't deal much damage with your spells and gain health from your spell-vamp.
This part is a little less straight forward, though pretty common sense.
I find that teleport and clarity are important to mages going mid. Since both summoner spells get a boost from the same mastery point, you may as well help their cooldown. Reducing callback time is a pretty big deal for solo mid, so taking a point in that can be pretty beneficial. Mana regen is pretty self explanatory for a mage, as is increased mana. By using a few points to increase your spell vamp AND lifesteal, you make Morgana even more capable of sustaining herself solo without having to return, which is great when your harass keeps making your opponent have to keep leaving combat to refresh. By taking a couple points to increase your gold passively gained, you make it easier to get middle and top tier items faster. Getting those items faster can really help you push harder. Finally, dropping your last point in experience gained will help you level faster, and get a slight edge on your opponent.
I don't focus as much on runes as I probably should. Mostly, you'll want to make sure you have a few to get that last 1% of cooldowns you lack from items or masteries. It's not much, but runes help fill the gaps like that. I also use runes to help increase AP, gold, and mana regen. I'll update this later when I have more time.
There's a few different ways you can go with summoner spells for Morgana. I usually go with Teleport and Clarity.
Teleport will let you bounce back into combat more quickly. This is especially useful mid, because if you have to go back for mana or hp, there's no one to hold the line except minions. Try as they might, they are no substitute for a champion with AOE harassment abilities.
Clarity is fundamental, because Morgana tends to burn through mana pretty quickly. Until you get a Chalice of Harmony, you'll need to recover mana somehow. Supplementing with a philosopher's stone is useful, but a quick boost is better early to mid game than slowly recovering.
Flash Most players seem to take flash, which is incredibly useful, I'll admit. However, with all the harassment you have, and the stopping power of Dark Binding, you really don't need this. If you feel like you can't live without it, drop teleport for it, not clarity.
Ghost Very useful for getting out of combat. It also has a shorter cooldown than teleport or flash, so a viable option. I still find that bouncing halfway across the field is more useful, but that's just me.
Ignite Definitely a useful spell. It helps finish off opponents like a poison. However, most of Morgana's spells have a much farther reach. I tend to avoid this spell, but is still an option.
Heal Do not pick this. With all the spell vamp and lifesteal that has been built up by this point, it's pretty useless except very early game, and even then you'd have to be a pretty abysmal player to need to heal that early.
Barrier Smite Clairvoyance Cleanse All of these are pretty much useless. They can give you a slight boost in certain situations, but for the most part I find that having other spells will help you do much better.