Spell vamp from Soul Siphon and magic damage from Tormented Soil allow Morgana to stay in her lane for a long time.
Use Dark Binding to check for enemies in the brush. The projectile stops moving and makes a distinctive sound when it hits something. While this method does work, it is best to save Dark Binding unless you have maxed cooldown reduction, as it is an essential ability for Morgana's combat and will leave you vulnerable if it is on cooldown.
With enough ability power, Dark Binding can deal a fair amount of damage as well as snare the opponent. The projectile moves fairly slowly, though, so always lead your target at any distance longer than close range.
Dark Binding is an exceptional zoning tool. If you're being chased, throw it at your enemies so you can get away. Even if you miss, it will cause enemy players to waste time moving around it.
Morgana's damage output is centered around a simple combo, the snare from Dark Binding followed by Tormented Soil. Soul Shackles can be used with this combo either after the combo to finish the enemy off or initially to slow the enemy and land Dark Binding easier.

For maximum damage, you can use Dark Binding followed immediately by Tormented Soil, then use Soul Shackles, followed by Tormented Soil then Dark Binding again, though this requires significant CDR . Black Shield can be used anywhere in that combo, ideally to prevent being stunned while using Soul Shackles so they don't get away.

Tormented Soil used on an enemy champion and following up immediately with Dark Binding to snare them in place is an excellent harassing tool in the early game. Low health champions can even be killed outright since Tormented Soil deals more damage to lower health champions and will hit for its maximum damage for every tick rather than grow in damage when the target is at much higher health.

Note that Tormented Soil deals a tick of damage immediately when placed on the ground. Use this knowledge to finish off an enemy fleeing with very low health.
When using Tormented Soil to harass an enemy, place it somewhat behind them. This is because they will naturally run backwards and doing so will cause them to take at least 2-3 ticks of damage. Eventually they will catch on and run to the side or forwards, use this to your advantage when casting Dark Binding.

Every tick of Tormented Soil on every champion procs the Tribute passive from Spellthief's Edge, a quick way to maximize gold generation.

If an opposing champion must be killed in a team fight, use Soul Shackles and Dark Binding on that champion - if it doesn't kill them outright, it will at least deal a significant amount of damage and leave them stunned.
Soul Shackles can be used to detect stealthed units and opponents in the bushes. The icon of Soul Shackles will become available only if an enemy champion is near you. It can be used to see where they are the moment you use it, but you can't keep track of them if they are moving.
Flashing into a group of enemies, using Soul Shackles and then using Zhonya's Hourglass is excellent to initiate team fights stun many enemies and come out with a fighting chance since they will likely be trying to scatter away.
Items that provide survivability make Morgana extremely difficult to kill in conjunction with Black Shield and Soul Shackles.
Black Shield is a vital ability to learn as it stops all forms of crowd control and also makes your target temporarily absorb magic damage.Shrewd use of Black Shield can determine the outcome of team fights.

Try to cast it pre-emptively on your teammates when they initiate, or as soon as you see them targeted by any spell or ability. If you can get the shield in before your teammate is hit, it just will absorb it.
Casting Black Shield on champions with channeled spells (e.g. Katarina and Fiddlesticks) will help keep their abilities from being interrupted.
Remember that Black Shield will only block Magic Damage, it will not block Physical Damage or related spells such as Ace in the Hole, Parrrley, Sonic Wave, or Volley.
It will still block the crowd control effects from Physical Spells and on-hit items, however.
If you play Morgana more offensively than supportive (at mid, for instance), it is best to go for higher levels of her other spells for farming unless you are against a heavily Crowd Control orientated team, as you will need damage to lane effectively.

Zhonya's Hourglass is one of the best items for Morgana; in addition to giving armor and massive ability power, the active is absolutely invaluable as it can be used while Soul Shackles is still casting and with proper positioning she can hit multiple enemies that would otherwise kill her while staying alive to finish them off after its over.

Athene's Unholy Grail allows for long-term mana sustain, reduced ability cooldowns, and magic resist.
Morgana is required to be in the center of a fight so that she can hit Soul Shackles on everyone. Thus, items such as Frozen Heart or Randuin's Omen are good items if you are unable to survive long enough to be effective in fights.

Soul Siphon

Morgana has Spell Vamp, healing herself whenever she deals damage with her spells.

Dark Binding

Cost: 50/60/70/80/90 Mana
Range: 1175

Morgana releases a sphere of dark magic. Upon contact with an enemy unit, the sphere will deal magic damage and force the unit to the ground for a period of time.

Fires a bolt of dark energy, rooting the first enemy hit for 2/2.25/2.5/2.75/3 seconds and dealing 80/135/190/245/300 (+90% Ability Power) magic damage.

Tormented Soil

Cost: 70/85/100/115/130 Mana
Range: 900

Infects an area with desecrated soil, causing enemy units who stand on the location to take continual damage.

Curses an area for 5 seconds. Enemies on the cursed ground are dealtfrom 24/38/52/66/80 (+22% Ability Power) to 36/57/78/99/120 (+33% Ability Power) magic damage every second based on the enemy's missing Health.

Black Shield

Cost: 50 Mana
Range: 750

Places a protective barrier around an allied champion, absorbing magical damage and disables until penetrated or the shield dissipates.

Shields an allied Champion for 5 seconds. The shield absorbs 95/160/225/290/355 (+70% Ability Power) magic damage and prevents disables until it breaks.

Soul Shackles

Cost: 100 Mana
Range: 625

Latches chains of energy onto nearby enemy champions, dealing initial damage to them and slowing their Movement Speed, and then echoing the pain a few seconds later and stunning them if they remain close to Morgana.

Dark chains latch onto nearby enemy Champions dealing 175/250/325 (+70% Ability Power) magic damage and slowing them by 20% for 3 seconds. After 3 seconds, they are dealt an additional 175/250/325 (+70% Ability Power) magic damage and are stunned for 1.5 seconds.Enemy Champions can break the chains by moving away from Morgana.

Lore

There is a world far away populated by graceful and beautiful winged beings gifted with immortality, where an ancient conflict still rages. Like so many conflicts, this war split families. One side proclaimed themselves as beings of perfect order and justice, fighting to unite the world under their law and strong central governance. Those that fought against them saw their kin as tyrants, creatures incapable of seeing the larger view, who would sacrifice individuality and freedom for the illusion of efficiency and safety. Morgana was one who fought against what she perceived as the tyranny of her kind, and for that she was branded ''fallen.'' Morgana was not innocent, having plumbed forgotten ways to gather forbidden might to become a powerful mistress of the black arts. This goal was driven by her obsession to defeat the general of the opposition's army - her sister, Kayle.

While the two were in fact birth-sisters, Kayle struck the first blow by disowning any filial connection when Morgana refused to join her cause. Eventually, Morgana grew in power enough to not only reach, but challenge Kayle. As the time approached when the two would meet in what could be their final conflict, Morgana was suddenly summoned to Valoran. At first, Morgana made a deal with the League's summoners to fight in exchange for greater power. With the advent of Kayle into the League, Morgana now willingly fights in the League of Legends for the privilege of being able to destroy her sister again, and again, and again. She lies in wait for the day the bonds of the Institute of War no longer hold her, and on that day she plans to destroy Kayle once and for all and return home a hero.

''There is no rest while Kayle's brand of tyranny still exists.''
-- Morgana

Quints
Quints of Ability Power I use this runes for early poking.

Glyphs
Glyph of Magic Resist This is standard, I always use this runes and it's good if they have ap support.

Seals
Seal of Armor I use this runes when I support because it's good if they have ad support and because I want to absorb ADC damage.

Marks
Marks of Armor Same as seals. It's good for survivability on bot.