Pros:
Great Farmer
Not as squishy as some other caster/support champions
Does quite a lot of damage
Your team will love you
Decent 1v1 against squishies (There are exceptions)
Cons:
Skillshots can be a pain
Not good at 1v1 against tanks and stronger dps (Kat for example)
Fairly slow movement speed
Veigar will grape you with his ultimate

This is my first build on Mobafire so I'm sorry if it doesn't meet your criteria because I haven't put pictures in all over the place to break up the blocks of text, but anyway, onto the build.
If you're new to playing Morgana or just looking for another build, I strongly suggest reading all of this guide, not just skipping to the parts and leaving others out. This build may not work if you only read a snipet.

Just as a disclaimer:

I generally play Morgana in mid so this build is just going with my experiences in mid, if it works for laning with a partner then great, if not, then I'm sorry this build didn't work out for you.
I'll continue to edit and update this guide after playing more games trying some more alternatives and a wider variety of teams, and of course after your feedback :D

What to buy

Doran's Ring - Great starting item.

Catalyst The Protector - Gives you a bit more survivability and lets you lane longer

Sorcerer's Shoes - Magic Pen and Move-Speed, should be the obvious choice

Rod of Ages - Survivability and some AP (You should have this at level 12-13)

Will of the Ancients - Relatively cheap, buffs your team with an AP and spell vamp aura.

Morello's Evil Tome - Mana regen, cooldown reduction and some more AP, just what you want.

Rylai's Crystal Scepter - Health and more AP, awesome. With a Rylai's you'll have around 3k health which is should be more than enough to survive any team fights

Sell your Doran's Ring To make space

Rabadon's Deathcap - Should push your AP up to around 500-600 and with this item you can easily shut down squishy champs.

A few alternate/situational items:

Zhonya's Hourglass: Armour, AP and an active/ability that makes you invulnerable, if you're getting focused I highly recommend this. Don't forget to bind it to a key to make it easier and faster to use.

Mejai's Soulstealer: If you're doing well at the start of a game go ahead, but your team should be the one getting most of the kills so I'd recommend the other items over this but of course if you have a thing for Mejai's Soul-stealer then don't let me stop you. If you are going to buy this item I'd suggest buying it in between boots and Rod of Ages.

Abyssal Sceptre: Taking a heap of magic damage? Grab this.

Deathfire Grasp: Fair amount of AP, Cooldown Reduction, Mana regen and very powerful active, a very good and valuable item if your enemies are tanky.

Frozen Heart: Is the opposition AD heavy? Consider grabbing this. Armour and an AS debuff for the opposing team will really help your own team and keep you alive longer.

Moonflair Spellblade: Cheap and has tenacity, not a bad item but if you're getting CC'd alot then I'd just get Mercury Treads

Void Staff: AP and magic pen, not a bad item but there are better.

Wards and Elixirs

Sight Ward
Vision Ward
Elixir of Brilliance
If you have spare gold at any time you recall buy an Elixir of Brilliance and Sight wards. Warding the two areas of bush on either side of mid will help you spot incoming ganks and will help with ganks of your own. If they have a stealth character buy Vision wards instead. Getting into the habit of placing wards and buying elixirs will make you better player and also give you an edge over the other team.

Greater Mark of Magic Penetration: Magic Pen - It comes in handy at all times during a game, especially at the start (1-6 mainly)

Greater Seal of Scaling Ability Power: More AP so you hit harder late game.

Greater Glyph of Scaling Ability Power: Glyphs of Force (AP per level), they give you a fair amount of AP start-mid, and they still help at the late game stage.

Greater Quintessence of Scaling Ability Power: So you hit harder faster.

As an caster you should go with the standard 9/0/21, shown in the mastery trees in this build,

Passive

Soul Siphon

Spell vamp

Not the greatest passive (35% spell vamp at its max) but if you purchase a Will of the Ancients (60% Spell vamp) or have a lot of AP you'll see the difference this makes after using (W) Tormented Shadow on a minion wave.

(Q)

Dark Binding

Max this ability first

- A skill-shot that snares the enemy and also deals a lot of damage. This ability will be your main way of taking out champs and helping out your team due to the snare. Dark Binding has a fairly high AP ratio so once you get your deathcap this will hit like a truck. You don't just have to use this ability offensively, you can also use it to snare an enemy chasing your allies, saving your ally and preventing the enemy team from receiving the kill and gold that comes with it.
To make Dark binding as effective as possible you need to learn to skill-shot well, mainly leading your target because missing is sadly not helpful to you or your team.

(W)

Tormented Shadow

Max this ability second

- An AoE that does damage over time and causes a debuff on enemies standing in it. Tormented Soil is an amazing farming ability, this will help you push mid constantly and get a fair amount of gold aswell. Although Tormented soil is great for farming,it also goes hand in hand with Dark binding (Q) when harassing and attacking champions. To be most effective when using the Dark Binding (Q) + Tormented Soil (W)combo you should put down your tormented soil first and then hit the enemy with Dark Binding: although this is hard to master it will just do that bit of extra damage on enemies.
If you are not completely confident in your ability to hit an enemy with dark binding, just go with the regular Dark Binding (Q) => Tormented Soil (W).

(E)

Black Shield

Max this ability third

- This is a shield that blocks magic damage only just so you know. Blocking magic damage is good and well, but the greatest thing about this ability is that it stops all disables... stuns, slows, suppresses, the whole lot.
Being immune to disables is an amazing ability; you can help a team mate get away, help yourself getting away, counter ultimates like Warwick's (Stops his ultimate in its tracks, he simply teleports to you.) and can simply be used to soak up some magic damage. Using it at the right time or even using it at all can make a huge difference in a team fight and helping team mates should be one of your highest priorities.

(R)

Soul Shackles

Max this ability WHEN POSSIBLE

- A good, fairly all around ultimate. Soul Shackles slows nearby champions, dealing instant damage and then tethering them to you, if you stay in range of champions tethered to you Soul Shackles does the same amount of damage again aswell as stunning them after 4 seconds.
This doesn't mean you should try and tower dive waiting for the second tick though.
I personally use this ability for the first burst of damage and the slow, but if its possible to stay in range for the second tick, do it. This ability will also make your team's life easier in a team fight, especially if the second tick goes off, stunning a few of the enemy's champions can easily turn the tide of a battle.
Besides being used as a damage, you could always use this ability for the slow in case you need to get away or help an ally get away.

In my honest opinion summoner spells are completely up to the player's preference, but if I had to choose for you I'd pick Teleport and Clarity.

Teleport: If you're in mid then this is definitely one of you're best friends, it helps you get back to farming minions and will get you back to your turret faster if it is being attacked. An all-round handy spell.

Clarity: Since I personally don't use mana regen runes, this is also one of my best friends, and will happen to be any of your mana hungry team mates' best friend, especially since you should have specced into improved clarity.

Flash; Most OP thing in the game at this very moment. This is your get out of jail free card and will save you from most situations. Flash could also be used to Flash => Soul Shacles => Zhonya's to initiate a team fight.

"I generally play Morgana in mid so this build is just going with my experiences in mid, if it works for laning with a partner then great, if not, then I'm sorry this build didn't work out for you.
I'll continue to edit and update this guide after playing more games trying some more alternatives and a wider variety of teams, and of course after your feedback :D"

Seeing as some people commented on Morgana's survivability: You really shouldn't be putting yourself in a position where you're being the one targeted, you should be at the back of your team trying to land your skillshot on the enemy team and throughout the whole game you should be buying Wards and Elixirs. Remeber: Save a life-Buy a ward.
Another point is that you have black shield to get away or stop at least some damage.
In addition to Black Shield you have your snare to root any pursuers in place.
If you're getting focused as Morgana get a Zhonya's Hourglass. If you're still dying because the other team focuses you even when you've got Zhonya's active on then your team should at least be winning team fights because your opposition is clearly a bunch of idiots.

Please rate and comment if you have any questions, quarrels or anything else, I'd really appreciate it. Kudos to those read this guide through to the end, thanks for reading and goodluck!