Round half to even is a mathematical heuristic that approximates an unbiased method of rounding.
Most of us learned to round off numbers to the nearest integer in a biased manner: if the trailing digit is 1-4, you round down, if it is 6-9 you round up; and if the trailing digit is a 5, we were taught to round this up also (this is called round half up). But this introduces a bias towards inflated numbers -- of the possible trailing digits, four are rounded down (1, 2, 3, 4), but five are rounded up (5, 6, 7, 8, 9).
One solution to this is to round half to even; when you are faced with a trailing five, you round to the nearest even number. In most cases, even and odd numbers are randomly distributed, so this should cause you to round 5 up half the time and down half the time, removing the bias.
This rule is used in many disciplines, and thus has many names. Names include unbiased rounding, convergent rounding, statistician's rounding, Dutch rounding, Gaussian rounding, odd–even rounding, and bankers' rounding. There are, of course, other ways of adjusting for unbiased rounding; perhaps most obvious is stochastic rounding, in which the decision to round up or down is chosen at random.