The Elder Scrolls V: Skyrim/First Lessons

When you arrive at the bridge leading to the College of Winterhold, Faralda will say that the bridge is dangerous and the gate will not open. To enter, you need to demonstrate to Faralda that you are able to cast spells.

Gateway test
If you can't use speech to convince her that you can succeed the test, she will ask you to cast one of several spells. The given spell will change, but as long as you cast it on the sigil next to her, she will permit you to pass.

The spell appears to be chosen randomly, although if you need to enter the college due to the main quest Elder Knowledge, you also have the option to demonstrate a shout. If you don't know a given spell, she will sell it for 30 gold.
 * Mage Light (84 magicka)
 * Conjure Flame Atronach (150 magicka)
 * Firebolt (41)
 * Fear (153)
 * Healing Hands (25/second)

If you don't have enough magicka to cast a spell, you will have to boost it using a potion of extra magicka, magic items with the fortify magicka enchantment, or when you improve the magicka stat as you gain levels.

Once you pass the test, she will bring you to the college.

Mirabelle Ervine
Report to Mirabelle Erving. She will give you some basic clothes more suitable for casters (basically college boots, novice hood and novice robe) to reduce the casting cost of some spells. She then takes you on the tour of the college, showing you your room where you can rest and store items.

She sends you to the main hall, to commence your first lesson.

Tolfdir
In the hall, you meet with the teacher Tolfdir, running a class with three apprentices. However, these apprentices are a little impatient, and want something a bit more practical.

If you do not know any wards, he will teach you the Lesser Ward spell. He then instructs you to stand at a spot, and use a ward to protect yourself from a magic attack. If done right, the spell will be absorbed and the damage negated.

He will announce an expedition for the apprentices as a good learning opportunity. This starts the quest ../Under Saarthal/.