The Eye of Magnus

Serving as the final quest in the College of Winterhold questline, it culminates in a battle against Ancano, with the Dragonborn becoming Arch-Mage upon emerging victorious.

Background
The Staff of Magnus has been recovered, and so it's time to assault the College of Winterhold and strike down Ancano.

Into the Eye of the Maelstrom
The College of Winterhold is now caught in an increasingly unstable maelstrom of magic - so much so that it has disappeared from the world map. The Staff of Magnus can be used to force clear the way across the courtyard and into the Hall of the Elements. There are likely to be one or more Magic Anomalies just inside the whirling force screen.

Upon arriving at the Hall of the Elements, Ancano is waiting, directing energy of some kind at the Eye of Magnus. He is well aware of the Dragonborn's presence. The mad elf spots them and Tolfdir, taunting and crowing that he's so far beyond any pitiful attempts at magic that he cannot be touched.

Unfortunately, this is true, as Tolfdir's aimed fireball dissipates harmlessly. The old mage has just enough time to tell the Dragonborn to bring out the Staff of Magnus and use it on the Eye before an enraged Ancano strikes him — and anyone else (such as followers) in the vicinity — with a Mass Paralysis spell, leaving the Dragonborn standing alone.

Ancano decides to finish the Dragonborn himself, opening the Eye of Magnus and tapping into its full and incredible power.

Note: It is possible to keep Ancano from paralyzing Tolfdir and any other follower by blocking the former's Mass Paralysis spell with a magic ward.

Defeating Ancano
Note that any followers cannot aid in the fight and will likely attack Tolfdir after the magic anomalies are defeated, so it will be impossible to complete the quest with one around.

While the Eye is open, Ancano is invincible. When the Eye is fully open, a casting from the Staff of Magnus absorbs its energy, and the Eye begins to close.

Once the Eye is closed, the Dragonborn can start attacking Ancano with effect. Eventually, the Eye reopens, and Ancano becomes invincible again. When the Eye is pried open a second time, several Magic Anomalies are also let loose. Although the Dragonborn must deal with them, their presence has the potential to be very useful, as the soul gems they drop can recharge the staff if it is out of charge. Additionally, the magic anomalies attack Ancano if the Dragonborn is not near him. This quest can be used as a way to farm filled soul gems for training in Enchanting.

The Dragonborn can use the chamber's topography to an advantage, keeping a healthy distance from Ancano, which allows for more time to maneuver around his spells. Standing behind one of the chamber's many pillars, or the Eye itself, will block the full force of Ancano's attacks. A very easy way to safely battle him is to stand in the Hall of the Elements behind the closed gate after Tolfdir is paralyzed. One can stay safely behind the gate to heal or take a break from the battle, or just to sit back and let a summoned follower do the work.

An alternative method, if the Dragonborn has the Sanguine Rose from "A Night to Remember", is to use it to summon a Dremora after Ancano casts his paralyzing spell. The Dremora will attack Ancano, leaving them free to use the Staff of Magnus on the Eye. The Dremora will make short work of him as soon as the protection of the Eye is removed.

Another way of quickly dispatching Ancano is to use high archery skill, have health damaging poison, and carry a powerful bow and arrows. With this combination, it is possible to use the Staff to close the Orb. Immediately after it starts to close, a poisoned bow with glass or above arrows can make short work of him.

One other method is to get a Staff of Paralysis (the Paralyze spell, and the level 100 archery perk will also work). Once the Eye is closed, the staff or spell can be used to paralyze Ancano. Then he can be killed easily.

The Nick of Time
With the hated High Elf now dispatched for good, the Dragonborn can return to Tolfdir. He is thankful that Ancano has been stopped, but has no idea how to dispel the gathering crescendo of power that the Eye of Magnus is exuding. Fortunately, Quaranir appears just in the nick of time.

Although he is as cryptic as ever, Quaranir tells the Dragonborn that the Psijic Order always believed in them, and their actions have proved their readiness to lead the College of Winterhold. Now that the Eye has grown unstable, it may not only destroy the College, but also the entire world.

Ancano's actions proved that this world is not ready for such power, so Quaranir prepares to secure the Eye. The Psijic Order will maintain the Eye for now, and he summons his brethren, Gelebros and Tandil, to perform a binding ritual. The monks and the Eye soon disappear from this world.

Aftermath
The quest's namesake relic is transported out of the Hall of the Elements by members of the Psijic Order. Tolfdir awards the Dragonborn the key to the Arch-Mage's Quarters as well as his robes.

After this, only the radiant, "Aftershock", and master-level ritual spell quests are left to complete for the college.

Achievements
Das Auge von Magnus El ojo de Magnus (Misión) L'Occhio di Magnus Око Магнуса (Квест)