Morgana is a weird mix of a support and an AP carry. She can help her team with her various CC and her unparalleled support move, Black Shield. When built right, she can also do insane amounts of damage. The only problem with building like this is that Morgana is very dependent on positioning and will be incredibly squishy with a straight AP build. In this build I try to get the best of everything: a decent amount of AP, 39% CDR, a couple of auras, and plenty of durability.

Clairvoyance: A must-have-one-per-team summoner spell. Definitely worth taking if you are your team's support.
Teleport: A good spell for both laning and preventing backdoors.
Either of these spells can be interchanged with Flash, Ignite, or Clarity, depending on needs and preferences.

9/0/21 pretty standard. Just be sure to grab masteries corresponding to your summoner spells if you choose different ones. Archaic Knowledge and Presence of the Master are must haves.

Getting 3 fast ranks of Tormented Shadow allows you to kill caster creeps in one spell. Grabbing Black Shield can be delayed if you aren't afraid of CC based jungle ganks and you aren't laning against a caster. Otherwise, prioritize:

Soul Shackles> Dark Binding> Tormented Shadow> Black Shield

Greater Mark of Magic Penetration: Best mark for any caster; no other option.
Greater Seal of Scaling Mana Regeneration: Morgana is pretty mana intensive so these help out.
Greater Glyph of Scaling Ability Power: Take these because the item build caps CDR. Alternatively, you can grab Greater Glyph of Scaling Cooldown Reduction and replace Frozen Heart with Zhonya's Hourglass. Do this if you have a teammate already grabbing Frozen Heart.
Greater Quintessence of Magic Penetration: These help out since you normally don't grab Sorcerer's Shoes. Quintessences are very flexible: you can grab Greater Quintessence of Health for low level survivability or Greater Quintessence of Movement Speed because those are always good.

Main Build:
Doran's Ring: Helps Morgana get off to a solid laning phase, which she is somewhat dependent on.
Boots: Getting Rod of Ages fast is important but you can't let yourself be outrun in your lane.
Rod of Ages: Basically has all the stats you could ever want at this point in the game.
Mercury's Treads: Boots that are good against most competitive teams as they have a lot of CC.
Soul Shroud: Really good support item that also boosts your survivability.
Glacial Shroud: Once you have a good health pool you need to start getting some armor and MR. This also gets you close to the CDR cap.
Abyssal Mask: Some much needed MR, a good damage boost, and the MR reduction helps your team as well.
Frozen Heart: Great aura, great armor, hit the CDR cap. Good item.
Rabadon's Deathcap: Adds more damage to the build as you have now pretty much covered all of your other other bases.

Other Good Items That You Might Use At Some Point:
Sapphire Crystal + 2 x Health Potion: You can start with these if you are confident in your laning ability. I would recommend Greater Quintessence of Health if you take these.
Sorcerer's Shoes: If the enemy team has little CC you can take these for increased damage. Ionian Boots of Lucidity could be decent to if you don't want to build all of the CDR items.
Aegis of the Legion: Surely someone else on your team would take this, but if they don't, it might be worth picking up.
Zhonya's Hourglass: A decent replacement for Frozen Heart but you should consider taking Greater Glyph of Scaling Cooldown Reduction if you do that. The active on this item is great.
Banshee's Veil: You're generally going to want to take Abyssal Mask, but this item still has a niche role against heavy CC.
Quicksilver Sash: An item every champion should consider if suppressions are killing you.
Guardian Angel: A good last item if you are getting focused in team fights. Turns you into a semi-tank.
Archangel's Staff: Standard for most Morgana builds; take it if you want more AP, possibly replacing Rod of Ages.
Haunting Guise: A solid mid game item that boosts health and damage.

Items That Seem Decent At First But Actually Aren't So Great:
Chalice of Harmony: Especially with runes, you shouldn't be having enough mana trouble to merit buying this item.
Will of the Ancients: I really don't understand why so many people love this on Morgana. It's a pretty underwhelming item with a less-than-great aura. There's really no need to stack more Spell Vamp when you already have some from your passive. It's not going to keep you alive in team fights, as you have long cooldowns. The main threat is being focused with heavy CC, and having more armor, MR, and health is going to stop that more than spell vamp.
Warmog's Armor: You can get Morgana nice and bulky while also supporting your team and increasing your damage; there' no need to build straight tank.

In general the items to build on Morgana are pretty straightforward, and she can use most AP and support items.