Approach the College of Winterhold, now caught in an increasingly unstable
maelstrom of magic. Equip the Staff of Magnus, and use it to force your way across the
courtyard and into the Hall of the Elements with Tolfdir at your side.
Tolfdir and Ancano will exchange dialogue briefly before Tolfdir launches a spell at him, but with no effect. Tolfdir is knocked down and remains incapacitated while Ancano attacks.
He will still have his protective barrier, but this can now be removed by using the Staff of Magnus on the Eye of Magnus. It is important to use the Staff for a sufficient amount of time on the Eye of Magnus - until it returns to a solid, non-glowing sphere – because the protective barrier still provides Ancano with near immunity to all forms of attack when the Eye of Magnus is only partially weakened.
When the glow surrounding Ancano dissipates he will be vulnerable to your attacks. During the fight he may approach the Eye and regain his protective barrier; if this happens you must remove it again immediately.
When the Eye separates into several hovering and rotating pieces, this means it is gaining strength, and this may cause Magic Anomalies to appear. Be sure to use the staff on the Eye when this happens, so as to drain more of its power and make it return to its original form for some time. The Magic Anomalies will yield soul gems when killed, which you can use to recharge the Staff of Magnus if it depletes at any point during the battle.
Keep on attacking Ancano when he is vulnerable and he will fall before you. If any magic anomalies do appear, they will attack him as well as you, so he might get killed that way.
When Ancano is defeated, several members of the Psijic Order will appear before you. They congratulate you for your efforts, and declare you worthy of being the new Arch-Mage of the College of Winterhold. They will also state that the Eye of Magnus is unstable, and too dangerous for this world, before vanishing in a glow of light, taking the Eye with them.
When you speak with Tolfdir, he will agree that with the old Arch-Mage and his immediate deputy dead, you are now the best candidate to be Arch-Mage. He will give you the key to the Arch-Mage’s Quarters as well as the Arch-Mage’s Robes.