In the nick of time, and amid much political drama, Congress passed the American Taxpayer Relief Act on New Year's Day—averting massive tax increases for nearly all earners that were slated to take effect Jan. 1.

Even so, millions of people soon will feel something less than relief from the new law.

While the top 1% of taxpayers will bear the biggest burden, many other families, affluent and poor, will pay more as well.

The most immediate change affects nearly all workers: Congress allowed a two-percentage-point cut for the employee portion of the Social Security tax to expire. As a result, each will owe up to $2,425 more in payroll tax this year than in 2012.

Beyond that, the new law's effects will be highly individualized—and in some cases highly painful.