The College of Winterhold is one of the 4 factions in Skyrim. They are the equivalent of the Mages' guild in Oblivion, located slightly north of Winterhold. Winterhold can be reached by using the wagon in front of any Holds' stables, the first one you should encounter is the one in Whiterun.
Before entering the college, the gatekeeper Faralda wishes to see a demonstration of magical powers. She requests to see a spell appropriate for the Dragonborn's level before gaining passage. These spells include:
Firebolt
Fear
Fury
Magelight
Healing Hands
Conjure Flame Atronach
Conjure Familiar
Fireball
Although she does not state it explicitly; some spells, e.g. Fear, must be cast at the emblem on the ground or she ignores it. If the spell she requests has not yet been learned, she offers to sell it for 30 gold Gold. The gate beyond the bridge does not open until the proper spell has been demonstrated for Faralda.