The quest "The Eye of Magnus" starts once you return to Winterhold after obtaining the Staff of Magnus.

Open your quest journal:
The Eye of Magnus
The Staff of Magnus has been recovered, and so it's time too assault the College of Winterhold and strike down Ancano

Reach the Hall of the Elements
Use the Staff of Magnus to enter the College of Winterhold

As you climb up to the college from town, you will encounter a force field. Use the Staff of Magnus to break through it by equipping it and shooting at the forcefield.

If you talk to Tolfdir on the way, he will tell you that you must hurry if you are going to defeat Ancano. Enter the Hall of the Elements, then the chamber with the Eye of Magnus. Your goal now is to defeat Ancano. Open the gate. Ancano will threaten you. Tolfdir shoots a fireball at Ancano, but it has no effect. Tolfdir tells you to use the staff on the eye. Ancano paralyzes him.

The eye opens, making Ancano invincible. Use your Magnus Staff on it to negate its effect, then strike Ancano. If the eye opens one more time making Ancano invincible for a second time, just use your Staff of Magnus on it again as necessary. Have soul gems ready to recharge the staff. Use pillars in the chamber as cover if you need to restore magicka.

When Ancano is defeated, the quest log will update:
Ancano has been defeated, but there is still a tremendous amount of energy coming out of the Eye of Magnus. I need to talk to Tolfdir about what to do next.

Talk to Tolfdir

Defeating Ancano should release Tolfdir from the paralysis spell. Go and speak to him. Ask him what to do now. He will tell you that whatever Ancano did to the eye doesn't seem to have stopped and he has no idea what to do. Quaranir shows up just in time. Talk to him. He will tell you they knew you would succeed and that you have proven yourself more then worthy to guide the College of Winterhold. Ask him how they knew you would be able to defeat Ancano. He tells you they have believed in you for a long time. Ask him what do we do now. He will tell you the eye has grown unstable and cannot remain here otherwise it could destroy the college and even the world. It must be secured, for Ancano's actions prove that the world is not ready for such thing. They  will safeguard it for now. You now have the opportunity to maintain your college and carry on with your lives and have their gratitude.

Some more Psijic Order monks appear, and they perform some ritual which makes the eye, and them, disappear. Tolfdir approaches you telling you that you have his gratitude. He says there is no one more serving to be Arch-Mage than you. He gives you the Arch-Mage's quarters key and the Arch-Mage's Robe. He says he will always be there if you need advice. The ArchMage's Robes are pretty useful, decreasing by 15% the cost to cast all spells, giving you 50 more magicka, and making magicka regenerate 100% faster. The key lets you access the quarters at any time, sleep on the bed there, store your items in there, as well as help yourself to magic potions and other magical items in the quarters.

"Ancano has been defeated. The Psijic Monks have removed the Eye of Magnus from the College, saying that the world is not ready for such thing. With the crisis over, I have been named the new Arch-Mage of the College"

Congratulations, this completes the main quests of the College of Winterhold and puts you as the Arch-Mage in charge!