What year did Skyrim begin?

The game of Skyrim starts on Morndas, the 17th of Last Seed, 4E201, at approximately 9:30.

If you don't know what Morndas or Last Seed are, I suggest you look at the answer to this question.

Skyrim starts at 4E 201, 200 years after the Third Era ended along with the Septim bloodline.

If you're playing on PC, you can type help gamedayspassed in the console to see how much time has passed in-game.