When you first come to the bridge leading to the College of Winterhold, you will come across Faralda. She will ask you to prove your magical ability by casting a spell. The required spell does vary, but she will always have it for sale for 30 gold (which is way cheaper than anywhere else in the game). Cast the spell and she will open the way for you inside the College. You can bypass the spell test by persuasion or also by claiming you are the dragonborn and performing a Thu'um demonstration