Morgana brings a lot to the bot lane. She has a root, a stun, a shield that prevents cc and good overall damage. She works very well with ADC's Sivir, Lucian and Caitlyn. She can poke well and assist ganks very well. She works wonders against cc heavy supports such as Leona, Taric and Annie, due to your black shield.
To deal damage as Morgana, you have your Binding and your Tormented Soil. Be aware that your cool downs are rather high, and can be taken advantage of by your opponents. If you land a Binding on the enemy ADC, you can either throw down your Tormented Soil to deal a little poke or go in with your ADC for the kill. This works best if your ADC can quickly react to your actions, so communicate!
If the enemy jungler comes to gank, shield your ADC or yourself (depending who is at risk) and bind the main threat (either damage threat or cc threat).
If you push you lane and/or take the turret, you can go roam mid lane to get a kill. Do this only if you know where their jungler is. Otherwise, stay bot with your ADC.
In team fights your main role is to protect your MVP, be that your ADC, top or mid. Use your shield on the initiating champ if they will need to go through a lot of cc to get to their target, your MVP so they can deal their damage, or someone who got caught. If someone jumps your MVP you can always use you binding or ult to kite them. Throw down you Tormented Soil in the middle of the enemy team for maximum damage and sustain.
Late game, you have to make picks with your Binding (3 second root at max rank) and protect your carries with your shield/kiting ability. If you get caught, you probably will die. However, if you get caught with your team next to you, you can use your ultimate and Zhonyas to do as much as possible.
Items are very important. If you follow my example of an end game build, you have 2 slows, 1 invulnerability (Zhonyas) and 1 cc removal + heal. This is my favorite build as it lets you create a lot of picks whilst also allowing you to kite the enemy team.

Thank you for reading. This is my first guide, so its not very detailed or pretty. I'm not a pro, but I really enjoy Morgana Support and wanted to spread my idea's. Thank you for reading! <3