Upon completion of Good Intentions, you will need to speak with Mirabelle Ervine concerning the Staff of Magnus. Ervine will reveal that researchers from a Cyrodiil mage council called the Synod were directed to a Dwemer ruin called Mzulft to find sources of magic.
You'll need to head for the ruins, located southwest of Kynesgrove in Eastmarch. Upon entrance, you'll need to speak with a dying researcher, Gavros, who will warn you about the Falmer who have occupied Mzulft. Acquire the key from his corpse and proceed further, first being wary of Dwemer animonculi guardians and then the dreaded Falmer themselves, including their pet sentries the Chaurus.
Make your way to the Aedrome and defeat a boss-level Falmer; be sure to loot the Focusing Crystal quest item from it, as you will need it very shortly. To gain entrance to the final portion of the dungeon, the Oculory, you'll need to acquire the key guarded by a levelled Dwemer animonculi; you may even be rudely introduced to the powerful Dwemer Centurion if you are high enough in level!
Defeat the guardian and search the chests for the Oculory key. Return to the door to unlock it, and meet the paranoid Imperial researcher Paratus Decimius. After a brief exchange, you'll need to have the Focusing Crystal in your possession before Decimius will help you any further; if you don't have it already, be sure to search all the Falmer you have defeated thus far in the Aedrome carefully to retrieve it.
With the Focusing Crystal, Decimius will lead you to the observatory in the Oculory itself. Decimius explains that the Crystal was needed to operate it; with the Crystal in place, you must help Decimius re-align it to get the Oculory to work again.
To do this, use Concentration Destruction spells like Flames and Frostbite to get the beams of light to hit the mirrors in the ceiling. Note that you will also need to adjust the position of the mirrors themselves using the buttons on the control panel in the upper level. You'll also find spell tomes up here that will teach you the spells needed to refocus the Crystal if you haven't already learned them.
Solving this puzzle basically requires you to find the right positions for both the mirrors and the lights emitting from the Crystal; use the spells to align the light beams to select positions. Then adjust the positions of the mirrors. You'll want to fiddle around with both in order to get all the beams of light to hit all the mirrors.
Note that you do not need to alternate between "expanding" and "contracting" the crystal with the fire/ice destruction spells to get the beams of light to move the way you need them to; you can get away with simply using one type of spell to get the light beams to move, as they can only be directed in a set number of positions and will "loop" through them all depending on how long you keep casting the concentration spell on the Crystal.
Once you successfully direct all the beams of light to hit the mirrors on the ceiling, the Oculory will activate and display a map of much of Tamriel with two key positions marked on Skyrim. While Decimius suspects foul play on your part due what he deems an unexpected and lackluster outcome, he will remain civil enough to explain that the positions refer to the College, and a massive dungeon complex known as Labyrinthian.
That's as big a lead as you can hope for in your search for the Staff of Magnus. In the meatime Decimius will jump to conclusions about the intentions of the College and gloat about figuring out its schemes. Feel free to do what you will with Decimius if you've had enough with his ranting, as he is no longer marked as an essential NPC at this point.
Attempt to leave the Oculory through the shortcut, and you'll once again be greeted by the appartion of Nerein from the Psijic Order who will warn you that something terrible has happened at the College, and instruct you to return there immediately to confront the situation.
And what a situation it is; while you were gone, Ancano has begun tapping into the powers of the Eye of Magnus and has set the College into chaos; help Arch-mage Savos and Mirabelle dispel the magical barrier which has surrounded the Hall of the Elements. Confront Ancano, who will simply summon a massive surge of magical energy to blow you all away, rendering you unconscious.
When you come to, you'll be instructed by an equally fatigued Mirabelle to seek the Archmage. This triggers the next main College quest, Containment.