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 Hold's stables, the first one you should encounter is the Whiterun one.
On your way to the College, you will be stopped by an npc to complete a test before entering. To complete it and net the achievement, you'll need to cast a random spell, which may or may not be from the skillset you chose to improve in the conversation. Most spells can be bought from the mage at Dragonsreach in Whiterun. He is located to the right of the Jarl in the main hall.
If you're not a mage centered character and depending on the spell you need to cast, you might have to put some levels into Magicka... you'll find out soon enough if you can cast the required spell or not. Alternatively, you can use potions to temporarily boost your magicka, so that you don't have to spend level-ups incase you want to keep them for another stat.
Needless to say that this place will be your haven for all sorts of Magic trainers and books, and mages should seek this place out as soon as possible.