Good Intentions is the 4th quest in College of Winterhold. You unlock this quest upon returning to Urag in the Arcanaeum after completing the quest "Hitting the Books"

Good Intentions
Urag has asked me to mention the Night of Tears book to Tolfdir

Objectives
Speak with Tolfdir

Find Tolfdir, most likely in the Hall of Elements. Speak to him and tell him Urag suggested you come see him. Tell him about "Night of Tears". Tolfdir will be distracted by the orb from Saarthal, and has some additional thoughts about it. Listen to what he has to say.

Ancano will come and interrupt. He has some important information. He needs you to come with him immediately. He will tell you someone from the Psijic order is here in the college and is asking to see you specifically. Follow Ancano. Enter the Arch-Mage's quarters. Speak with Quaranir. Ask him what's going on. He will tell you about the Eye of Magnus. The longer it remains in the college, the more dangerous the situation becomes. He will tell you to seek out the Augur of Dunlain within the college.

If you look at your quest journal, it now says
"I've been led to the Arch-Mage's quarters by Ancano, who was insistent that I speak to someone claiming to be from the Psijic Order. The monk, named Quaranir, has informed me that the "Eye of Magnus" is dangerous and that to learn how to deal with it, I need to speak with the Augur of Dunlain within the college. I should ask my fellow members where to find him"

Your objective is to find the Augur of Dunlain. Go to the Hall of the Elements. If you ask J'zargo about the Augur, he will tell you that Tolfdir will probably know. Onmund will also recommend you seek Tolfdir. Locate Tolfdir (Search for him in the Hall of Attainment). Inquire about the Augur. He will tell you the Augurn is down in the Midden underneath the college.

Open the local map and locate the entrance to the Midden. Find the grate on the floor and enter the Midden. Descend down the tunnels, fighting some Draugr on the way. Go through a wooden door and enter the Midden Dark. Cross an icy bridge and enter through a door. Activate the Augur of Dunlain. He will tell you your efforts are in vain, it has already begun.  He will tell you that you blindly follow the path to your own destruction. The Thalmor was there too. The one who calls himself Ancano. He also seeks information about the eye. You must find that what you need. You wish to know more about the eye of Magnus and avoid the disaster. For that you require the Staff of Magnus. He tells you to take this knowledge to the Arch-Mage.

Your objective now is to report to Savos Aren. Go back towards the bridge, drop down, and exit to Skyrim. Fast travel to the College of Winterhold and find the Archmage in the Hall of the Elements. Talk to him and tell him you have important information for him. Tell him you need to find the Staff of the Magnus. It's connected to the orb we found. You know of this because you spoke with the Augur of Dunlain. He will be impressed with your initiative. He wants you to follow up on this. He recalls Mirabelle mentioning something about the staff and recommends you check with her.

He will reward you with a Mage's Circlet which increases Magicka by 30 points. This completes the quest "Good Intentions" and unlocks the next College of Winterhold quest, "Revealing the Unseen"