What does it mean when someone says he is from the "Class of 2001"?

Solution 1:

The graduating class of 2001.

If it referred to the year he entered school (this can refer to US high schools as well as colleges), it would be "Entering class of 2001" or "Freshman class of 2001".

But nobody uses those forms for people who've graduated, or at all, really; unless they're trying to raise money.

Solution 2:

No matter when your graduation is, in years ahead, now, or years ago, in the US, a member of the 'class of 2012' is someone who has, is now, or will graduate in the year 2012.

Americans born this year will likely graduate from a university in the 'class of 2034', the seniors graduating this weekend are in the 'class of 2012', and at 25 anniversary class reunion in a month is for 'the class of 1987', the year of their graduation. Incoming freshmen in the fall are 'class of 2016' (four year college).