Example: Half of the pay period is in December 2014 and half of the pay period is in January 2015. Which tax table will Payroll use? The table for 2014 or 2015?

Payroll will use the Tax table for 2015 because the period end date actually falls in 2015.  The IRS rules are centered around when wages are actually paid (not earned), so any wages paid out in January, regardless of previous year overlap, will use 2015 taxes when calculated. In short, the employee isn't receiving any of the money until the 2015 year, thus all income is in the 2015 year and falls under 2015 taxes.