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You'll receive this quest when you approach Faralda on the bridge leading to the College of Winterhold.
In order to get past Faralda, you'll need to prove to her that you have some aptitude with magic. Faralda will ask you to cast one of five different apprentice spells (including Magelight and Conjure Flame Atronach) on the seal behind her, and if you don't know the spell, then she'll teach it to you for 30 gold. This is a pretty cheap way to learn a spell, so you might want to save before talking to Faralda to make sure that you get a spell you want.
Note: You might also be able to persuade Faralda to let you pass, and, depending on where you are with other quests, she might let you pass because you're dragonborn.
Once satisfied, Faralda will escort you across the bridge, and she'll direct you to report to Master Wizard Mirabelle Ervine, who you'll find in the courtyard (arguing with a Thalmor mage named Ancano). When you talk to Mirabelle, she'll donate a set of mage clothes to you --
Novice Hood, and
Novice Robes of Destruction -- and she'll give you a short tour of the college (mostly just to show you where your room is; feel free to loot anything that you find there).
After the tour, you'll need to enter the Hall of the Elements, where Tolfdir will begin a lecture about the dangers of magic and the usefulness of wards. If you don't know a ward spell, then Tolfdir will teach you one for free. Tolfdir will then ask you to help him demonstrate the ward, which means you'll have to stand on one of the seals in the room and keep your ward up until he casts a Firebolt spell at you. If you're not much of a mage, then you might have trouble maintaining your ward, so it's a good idea to have some magicka potions available just in case.
After the demonstration, Tolfdir will announce that the college is excavating the ruins of Saarthal nearby, which he considers to be a wonderful learning opportunity for new students such as yourself. Tolfdir will then ask you to meet him there. This request will end the current quest and trigger the next quest in the line,