You will often see a statistical test or a hypothesis referred to as one-tailed or two-tailed. A one-tailed hypothesis is simply one that specifies the direction of a difference or correlation, while a two-tailed hypothesis is one that does not.
For example, if we correlate income with years of schooling we might hypothesize that years of schooling tend to increase with income. That is a one-tailed hypothesis because it specifies that the correlation must be positive.
On the other hand, if we were correlating people's heights with their income (it's been done), we might have no good reason for expecting that the correlation would be positive (income increasing with height) or negative (income decreasing with height). We might just want to find out if there were any relationship at all, and that's a two-tailed hypothesis.
The practical significance of this distinction is that you can use a smaller sample to test a one-tailed hypothesis. The practical significance of using a smaller sample, of course, is that it often reduces your costs.
The terms one-tailed and two-tailed, by the way, come from the mathematical details of statistical testing (they refer to the tails or ends of the sampling distribution). The terms directional and non-directional are also used, although probably not often enough.
(2000, John FitzGerald)
