A non-directional hypothesis, in statistics, is a hypothesis used to prove (or disprove) that changing one variable has an effect on another variable. It does not ask whether the effect is positive or negative - see directional hypothesis for that kind of thing. Instead, let's say we were trying to see whether or not a certain drug affects people's moods. Our non-directional hypothesis set would be:
H0: Drug X does not affect your mood.
HA: Drug X affects your mood.
Generally when dealing with hypotheses testing in statistics we assume that the null hypothesis is true - that is, that there is no correlation between variable 1 and variable 2. We then take our sufficiently large samples and collect our data about the drug, we compare our expected value of mood swings among our patients with the actual value. Using complicated things like z-tables and formulas to account for sampling error, we can determine whether we should reject our null hypothesis (and say that the drug does affect mood) or accept it (and say the drug does not affect mood.) Notice that we don't care whether the drug affects your mood positively or negatively - merely that it affects mood at all.
In general, non-directional hypotheses are highlighted by the words "influence", "change", "alter", "affect", and the like. In political science, non-directional hypotheses can be important in weighting various issues among the people. For example, if President Bush ran a survey suggesting that only 20% of the people in the country considered the economy "an important issue" in determining how they vote for President, he might be less likely to spend time improving the economy; alternatively, if 90% of the people said they considered it an important issue, he might just well spend the next month or two focusing purely on it. Issue-weighting can affect entire campaign agendas, which is where us poli sci majors make our bread and butter.
Other places where non-directional hypotheses have a lot of power include the aforementioned pharmaceutical research; in economics when studying consumer confidence; and even in such hip things as information theory, where things such as data sources and decentralized dataflow are all analyzed with non-directional hypotheses for analysis and improvement.