Skyrim Secondaries- College of Winterhold: You're a Wizard, Dovahkiin

The College of Winterhold is renouned throughout Skyrim as the place to go if you want to learn about magic- regardless of what kind of magic it is that you seek.  This immense building is pretty much the only point of interest in Winterhold itself- whether this is recent or has been the way of things for some time isn't entirely clear.  The smashed-open houses and ruinous state of Winterhold make it clear that the city used to be more populous and popular, but how recently that was isn't entirely clear.

Either way, you'll likely have had the college mentioned to you multiple times as you travel around Skyrim, and that's bound to instill at least a little curiosity in any wanderer.  Of course, that doesn't necessarily mean you're going to get to satisfy your curiosity.  Indeed, your initial entrance to the college is blocked by one of the current students.

Faralda is one of the High Elves, or Altmer, that study in the College of Winterhold, and she is very wary of newcomers to the area seeking entrance to the school.  She stops you before you can even cross the bridge to the castle that holds the college and warns you not to enter if you try simply walking past her.  Indeed, without her help you can't even get in, as the gates are closed tight.

When you talk to her, she will tell you both what the college is and why it is that she waits outside the front entrance to the place.  She will then ask why you are here- you can respond by telling her what it is that you want, the options you are given depending on what magics you have the highest skill rating in and which schools you know the most spells from.  You can also offer that you 'just wanted to see what it looks like inside', which gets a bit of amusement from her.

Whatever your answer, she proposes a test for you- you can attempt to Persuade her that you will pass the test anyways, or you can agree- if you agree, she will request that you cast a spell on the sigil that she stands besides.  Which spell she requests depends on your earlier answer and your magical skill, and is usually a second-rank spell of the school.  If you do not have the spell, then let her know and she will give you a book so that you may learn it.  You may also get the option, provided you're far enough along in the main questline, to ask if she would grant entry to the Dragonborn- this will require that you demonstrate with a Thu'um cast at the seal.  Take care you don't use a damaging spell or Shout with a large area, as catching Faralda in it's effect will prompt a fight with her and prevent you ever gaining membership in the College.